On Quantum Theory

Eur. Phys. J. D (2013) DOI: 10.1140/epjd/e2013-40486-5 THE EUROPEAN PHYSICAL JOURNAL D Colloquium

Berthold-Georg Englerta Centre for Quantum Technologies and Department of Physics, National University of Singapore, Singapore

Received 9 August 2013 / Received in ﬁnal form 24 August 2013 Published online (Inserted Later) – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2013

Abstract. Quantum theory is a well-deﬁned local theory with a clear interpretation. No “measurement problem” or any other foundational matters are waiting to be settled.

Quantum theory had essentially taken its final shape by 1 Physical theories: phenomena, formalism, the end of the 1920s and, in the more than eighty years interpretation; preexisting concepts since then, has been spectacularly successful and reliable – there is no experimental fact, not a single one, that con- tradicts a quantum-theoretical prediction. Yet, there is a In physics, a theory has three defining constituents: the steady stream of publications that are motivated by al- physical phenomena, the mathematical formalism, and the leged fundamental problems: We are told that quantum interpretation. The phenomena are the empirical evidence theory is ill-defined, that its interpretation is unclear, that about physical objects gathered by passive observation, it is nonlocal, that there is an unresolved “measurement typical for astronomy and meteorology, or by active ex- problem”, and so forth1. perimentation in the laboratory. The formalism provides the adequate mathematical description of the phenomena It may, therefore, be worth reviewing what quantum and enables the physicist to make precise quantitative pre- theory is and what it is about. While it is neither possible dictions about the results of future experiments. The in- nor desirable to cover the ground exhaustingly, we shall terpretation is the link between the formalism and the deal with the central issues and answer questions such as phenomena. these: Each physical theory relies on preexisting concepts, which reflect elementary experiences and without which the theory could not be stated. In Newton’s classical me- – Is quantum theory well defined? chanics [2], these concepts include mass and force, and the – Is the interpretation of quantum theory clear? theory deals with the motion of massive bodies under the – Is quantum theory local? influence of forces. Classical mechanics cannot provide any – Is quantum evolution reversible? insight why there are masses and forces. – Do wave functions collapse? Likewise, electric charge is a preexisting concept in –Isthereinstantaction at a distance? – Where is Heisenberg’s cut? Maxwell’s electromagnetism [3], which accounts for the forces exerted by the electromagnetic field on charged bod- –IsSchrödinger’s cat half dead and half alive? ies and how, in turn, the charges and their currents give – Is there a “measurement problem”? rise to the electromagnetic field. Maxwell’s theory cannot tell us why there are electricity and magnetism or why there are charges and fields, nor is this the purpose of the

theory. In sincere gratitude for many instructive discussions, I ded- One preexisting concept of quantum theory is the icate this essay to Professor Rudolf Haag on the occasion of his event, such as the emission of a photon by an atom, the ra- 90th birthday. a dioactive decay of a nucleus, or the ionization of a molecule e-mail: [email protected] 1 in a bubble chamber. The formalism of quantum theory No references are given here or later; why point a finger at a particular representative of one or the other community? has the power to predict the probabilities that the events The pertinent essays in the 2009 Compendium of Quantum occur, whereby Born’s rule [4] is the link between formal- Physics [1] cite the relevant papers. This no-finger-pointing ism and phenomenon. But an answer to the question Why policy applies throughout the chapter. In particular, we note are there events? cannot be given by quantum theory. that the quotes in (53), (60), (62), and (63) are not made up. In addition, there are preexisting concepts that are They are actual quotes from existing sources. The sources are common to all physical theories: the three-dimensional irrelevant, however, the quotes are representatives of many sim- space in which the natural phenomena happen, and the ilar statements. time whose flow distinguishes the past from the future. Page 2 of 16

We must also acknowledge a fundamental philosophical the probability that the molecule gets energetically ex- concept of science – the conviction that it is possible, with cited without being ionized and subsequently emits one or means accessible to the human mind, to give a systematic more photons, and the probability that the molecule re- account of the natural phenomena. mains unaffected by the charged particle. Whether the ionization event happens or not, we cannot predict – the principle of random realization (Haag in [9]) ensures that the events do happen in accordance with their probabili- 2Events ties of occurrence. Born’s rule tells us these probabilities – the prob- The formalism of quantum theory can be used to predict ability for the ionization event, the probability for the probabilities – probabilities for events. In the classic Stern- 2 photon-emission event, and the no-event probability for Gerlach experiment [5] , the silver atom eventually hits the molecule remaining unaffected. For that to have any the glass plate, and we can calculate the probability that meaning, the existence of events must be accepted in the the next atom will land in a certain region. The hitting first place. In this sense, then, the event is a preexist- of the glass plate is localized in space and time and has ing concept of quantum theory. We cannot formulate the an element of irreversibility, videlicet the emission of long- theory without this concept. wavelength photons during the violent deceleration of the atom, which photons are irretrievably lost. Another example of an event is the radioactive α-decay of a nucleus, which is invariably accompanied by the bremsstrahlung 3 Measurements, Born’s rule associated with the escaping α particle and the two now- superfluous electrons that are left behind. Yet another ex- 3.1 An example ample is the absorption of a photon by a semiconductor detector. These examples illustrate the defining features Consider the set-up sketched in Figure 1 where photons of an event. are incident on a half-transparent mirror that serves as First, an event is well localized in space and time. a symmetric beam splitter (BS). Whether transmitted or Haag [8] emphasized, and rightly so, that short-range in- reflected, the photons then encounter a polarizing beam teractions are crucial for the localization of an event and splitter (PBS) that reflects vertically (V) polarized pho- that events are linked by particles, so that a causal history tons and transmits horizontally (H) polarized photons. For evolves – think, for instance, of the sequence of ionization simplicity, we assume that the BS is symmetric for all po- events in a bubble chamber. By their spatial-temporal re- larizations, so that the next photon to arrive has an equal lations, the events give meaning to the space-time struc- chance of being reflected or transmitted at the BS, irre- ture, the great stage for all natural phenomena. spective of its polarization. The photons that are reflected Second, an event is irreversible, it leaves a document pass through a quarter-wave plate (QWP), which converts behind, a definite trace. It has often been emphasized that right-handed (R) circular polarization to vertical polariza- an amplification is necessary to bring the event to the tion and left-handed (L) polarization to horizontal polar- attention of the experimenter and that the amplification ization. The four detectors D1,...,D4 in the output ports is an irreversible process; this is true, but it downplays (or of the PBSs, therefore, distinguish between H and V polar- even ignores) the irreversibility of the event itself, which ization or between R and L polarization of the incoming is independent of human observation. If you don’t have an photon. event in the first place, there is nothing you can amplify. The construction of the single-photon detector for a In the bubble chamber situation above, the amplification particular outcome is largely irrelevant, except that its turns the sequence of ionization events into a string of efficiency depends on which physical process is used for bubbles that reveal the track of the charged particle to absorbing the photon and how the absorption event is the human eye. amplified to finally make the detector “click”. With pho- Third, events are randomly realized. Before an event tons arriving one by one, separated in time by more than occurs, there are usually several, perhaps many, different the dead-time of the detectors, the measurement result events that could happen, each of them with a certain consists of a sequence of detector clicks. Quantum theory probability of occurrence. Eventually, a definite one of the cannot predict which sequence will be recorded for the possible events will be realized. We cannot predict which next, say, one hundred photons but, for the next photon one will be the case – because this is unknowable, not to arrive, it tells us reliably the probability of clicking for becausewearemissingacrucial piece of information. each detector, provided we know the polarization of this Quantum theory can be very helpful in understand- incoming photon. ing quite a lot of the details of, say, the ionization of a Which one of the four detectors will be triggered by the molecule by a charge that passes by. We can calculate, next photon, which one of the four possible events will be with reasonable accuracy, the probability of ionization, the actual one, this is not just unknown, it is unknowable. One of the events will be randomly realized. 2 The accounts by Friedrich and Herschbach [6] and by That quantum theory cannot predict which detector Bernstein [7] of this experiment’s history are recommended will click for the next photon, does not imply that quan- reading. tum theory is incomplete. There is nothing missing; it Page 3 of 16

.....BS PBS...... conventions: ...... input ...... ...... 1 H ...... D , port ...... 1 0 ...... |V , |H , . . = = ...... 0 1 ...... . 2 V . D , √1 1 √1 i . |R = , |L = . (1) . i 1 ...... 2 2 ...... QWP ...... Accordingly, the statistical operator for the polarization ...... R ...... D3, PBS ...... degree of freedom for the incoming photon is represented . . . . by a 2 × 2matrix, ...... D4, L ρ ρ V| ρ ρ ρ |V|H VV VH VV VH , = ρ ρ H| = ρ ρ (2) Fig. 1. A four-outcome measurement of photon polarization. HV HH HV HH An incident photon has equal chance of transmission or reflec- BS tion at the beam splitter ( ). The polarizing beam splitters and the probability that detector D1 will click for the next (PBS) reflect vertically polarized photons and transmit hori- photon is zontally polarized photons. Photons transmitted at the BS are probed for horizontal (H)andvertical(V) polarization and de- 1 1 p1 = η1ρ = η1H|ρ|H =tr{ρΠ1} (3) tected by detectors D1 and D2, respectively. The combination 2 HH 2 of quarter-wave plate (QWP)andPBS probes the photons reflected at the BS for right-handed (R) or left-handed (L) circular with the probability operator polarization, with respective detectors D3 and D4. 1 1 00 Π1 = |H η1H| = η1 . (4) is not possible to add further elements to the quantum- 2 2 01 theoretical formalism or its interpretation and then have 1 the power of making such predictions. There are no con- Here, the factor of 2 is the transmission probability of the sistent modifications of quantum theory that will achieve BS and η1 is the detection efficiency of D1, the conditional this without getting wrong predictions in other situations. probability that a click is triggered by an incident photon. The question Can Quantum-Mechanical Description of Similarly, the probability operators for the other three Physical Reality be Considered Complete? that Einstein, detectors are5 Podolsky, and Rosen asked in 1935 [10]3 has a clear answer: Yes, quantum theory gives a complete description of 1 1 10 Π2 = |V η2V| = η2 , 4 2 2 00 the phenomena. Bohr’s reply [12,13] hit the mark. IntheexperimentofFigure1,itisinfactimpossible 1 1 1 −i to prepare the photon such that a particular detector will Π3 = |R η3R| = η3 , 2 4 i1 certainly click. Yes, we can make sure that one of the de- H 1 1 1i tectors will not click – -polarized photons will never give Π |L η L| η . 4 = 4 = 4 − (5) risetoaclickofD2, for instance, but then the photon 2 4 i1 1 1 1 has probabilities of 2 , 4 , 4 of entering one of the other three detectors. Of course, what we have here is just an These need to be supplemented by the probability op- example of the fundamental probabilistic nature of quan- erator Π0 for the possibility that the photon escapes tum processes – an illustration of the principle of random detection (no click), realization at work. Π − Π Π Π Π , It should be noted that the reflection or transmission 0 =1 ( 1 + 2 + 3 + 4) (6) of the photon at the BS does not constitute an event. The to make up the probability-operator measurement photon needs a short-range interaction with a partner to 6 give rise to an event. If no such partner is available, we (POM )fortheset-upofFigure1. With the probability operators Πk at hand, the could complete a Mach-Zehnder interferometer by adding p mirrors and another BS, after which it would be unknow- respective probabilities k are obtained by Born’s rule, able if the exiting photon was reflected or transmitted at p {ρΠ }. the BSs. k =tr k (7) 5 For simplicity, we ignore here and in (3) the possibility of “dark counts” – spontaneous detector clicks not triggered by 3.2 Mathematical description an incident photon. 6 POM, with its emphasis on “probability” and “measure- We represent the kets for the polarization state of a pho- ment”, is preferable quantum physics jargon. The mathemati- ton by two-component columns, for which we use these cal term POVM (positive-operator-valued measure) makes reference to measure theory. Note that “theory” here identifies 3 Reprinted in [11]. a mathematical discipline and does not have the meaning of 4 Reference [13] reprinted, with pages in wrong order, in [11]. Section 1. Page 4 of 16

...... no click ...... therefore, use a different statistical operator and arrive at ...... different values of the probabilities pk. Yet, her predic- ...... D1 ...... D2 tions and ours, though different, will both be correct and ...... D3 verifiable. . . ρ . . input ...... measurement ...... port . apparatus ...... 4 Statistical operators ...... DK ...... 4.1 Mixtures, blends, and as-if realities Fig. 2. A generic measurement. A physical object (photon, electron,atom,molecule,...),whoserelevantquantumdegrees The general form of a statistical operator for the polariza- of freedom are described by the statistical operator ρ,enters tion degree of freedom of a photon is given in (2), where the measurement apparatus and triggers a click of one of the the matrix elements are restricted by K detectors D1,D2,...,DK , or it escapes detection (no click). ∗ ρ ρ A, B H, V, AB = BA for = ρ ρ , ρ ρ ≥ ρ ρ . VV + HH =1 and VV HH VH HV (11) As already noted in Section 1 above, Born’s rule is a defining element of the interpretation of quantum theory, it Let us consider a ρ with a positive nonmaximal links symbols of the mathematical formalism with the phe- determinant, nomenology: The statistical operator, which summarizes 1 what we know about the polarization state of the incoming > det {ρ} = ρ ρ − ρ ρ > 0, (12) photon, and the probability operator, which represents the 4 VV HH VH HV apparatus, with the probability of a detector click, which ρ2 ρ ρ we can determine in an experiment by observing suffi- so that the polarization state is impure ( = )and has ciently many repetitions of the situation “single incident the two positive eigenvalues photon triggers a detector click”. r1 1 1 = ± 1 − 4det{ρ} , 1 >r1 >r2 > 0. (13) r2 2 2 3.3 Generic measurements The rank-one projectors to the respective eigenspaces are The particular example of Figure 1 illustrates the general measurement scenario. Let us briefly recall the generic properties of a measurement in the quantum realm, as de- ρ − r2 r1 − ρ |r1r1| = and |r2r2| = , (14) picted in Figure 2. The relevant degrees of freedom of the r1 − r2 r1 − r2 quantum system are described by the statistical operator ρ,andtheK outcomes of the POM have probability and ρ |r r r | |r r r | operators Π1, Π2, ..., ΠK . = 1 1 1 + 2 2 2 (15) The statistical operator is positive and normalized to is the spectral decomposition of ρ. unit trace, If ρ refers to a large ensemble of photons, we could say ρ ≥ , {ρ} , 0 tr =1 (8) that it is as if ρ came about by having fractions r1 and and the probability operators are positive and have unit r2 of photons in the polarization states associated with sum, kets |r1 and |r2, respectively. Speaking of the one pho- K ton of interest, regarded as the representative of a Gibbs Πk ≥ 0, Πk =1, (9) ensemble of photons, it is as if that photon had respective k=0 chances r1 and r2 of being of the |r1 or |r2 kind. This is but one of very many as-if realities for the one statistical where we include Π0, the probability operator for the “no operator ρ. For example, we can also write click” case. The probability that the kth detector clicks is given by Born’s rule (7), which also applies for the no- 1 1 p ρ = |r+ r+| + |r− r−| (16) click probability 0. As a basic check of consistency, one 2 2 can verify without effort that the probabilities are positive and have unit sum, with √ √ |r± = |r1 r1 ±|r2 r2. (17) K pk ≥ 0, pk =1. (10) Now it is as if the photon had equal chances of being in |r |r k=0 the polarization states for kets + and − . Hereby, each choice for the relative phase between |r1 and |r2,when The probabilities pk are conditional probabilities – they factoring the projectors in (14) into ket-bra products, gives are conditioned on what we know about the physical sys- different r± kets in (17) and, therefore, different projectors tem, which information is encoded in the statistical opera- on the right-hand side of (16) and a different as-if reality to tor that we use for our probabilistic predictions. Someone go with them. Any convex sum of (15) and (16) identifies else may have different knowledge about the system and, yet another as-if reality. Page 5 of 16

We have the same mixed state ρ,thesamemixture,and a larger physical system in a pure state. Now, it is already different blends for it7, each of them offering a different quite challenging to acquire sufficient information about a as-if reality. All that matters is the mixture, not how it few degrees of freedom to justify, as a good approximation, is blended as a convex sum of pure states. Born’s rule (7) the use of a rank-one projector as the statistical operator. makes no reference to the blend and its as-if reality, it only How, then, can we have a pure state for the unknown de- cares about the mixture ρ. There is no way of distinguish- grees of freedom of a fictitious larger system that exists ing between the different blends of (15) and (16) or of their only in our fantasy? Of course, we cannot. many convex sums; no experiment can tell whether this as- It appears that sloppy language is one source of this if reality or that one is “what really is the case”. Attempts misconception. One often speaks of “the wave function at discriminating between blends founder on fundamental of the electron” or “the statistical operator of the pho- principles, such as Einsteinian causality [15,16]. All ways ton” rather than “our wave function for the electron” of blending a mixture are equally good, all as-if realities or “our statistical operator for the photon”. The precise are equally virtual. In particular, the spectral decomposi- “our . . . for” phrases leave no doubt that we are describing tion in (15) and the as-if reality associated with it have the electron by this wave function or the photon by that no special significance. statistical operator, whereas the sloppy “the . . . of” word- Of course, these observations about mixtures, blends, ings wrongly suggest an independent existence of the wave and as-if realities – here illustrated in the simple context function or the statistical operator. When this suggestion of photon polarization – carry over to more general situa- is taken seriously, the church gates stand wide open. tions. Whenever the statistical operator is not a rank-one Is there not the one largest system, the universe, with projector, there will be a plethora of blends for that mix- the physical system of interest as a small part of it? Yes, ture and a corresponding abundance of as-if realities. there is the universe, but no-one has a wave function for the universe, and there is no wave function of the universe.

4.2 Purification 5Evolution A mixture that is blended from J rank-one ingredients, 5.1 Equations of motion J J √ √ ρ = |ajwj aj| = |aj wj δj,j wj aj | (18) In Born’s rule (7), both the statistical operator ρ and j=1 j,j=1 the probability operator Πk are functions of the dynamical variables – collectively symbolized by Z(t)– w > w with positive weights of unit sum ( j 0, j j =1), with, possibly, a parametric time dependence as well. All can be written as the partial trace of a rank-one projector such operator-valued functions evolve in accordance with of a bipartite system, Heisenberg’s equation of motion, ρ =tr2 || (19) d ∂ 1 f Z(t),t = f Z(t),t + f Z(t),t ,H Z(t),t , dt ∂t i with (21) J √ the quantum analog of Hamilton’s equation of motion for |= |aj bj wj, (20) phase-space functions. In (21), H Z(t),t is the Hamilton j=1 operator, itself a function of this kind, and the commu- tator term accounts for the dynamical time dependence b b |b δ where the kets are orthonormal, j j = j,j .Since f Z t ,t Π H the blend that we start with in (18) is not unique and any of ( ) .Whereas k and may or may not have orthonormal set of b kets of any auxiliary second system a parametric time dependence, the parametric time dependence of the statistical operator compensates for its will serve the purpose, there are many different ket-bras ρ ||that one can use in (19) equally well. dynamical time dependence because is constant, In particular situations, such as the one discussed in d Section 7, we are really dealing with a bipartite system in ρ Z(t),t =0, dt a pure state (or nearly so), and then a physically meaning- ∂ 1 ful ket |may be available. In the generic case, however, ρ Z t ,t H Z t ,t ,ρ Z t ,t . ∂t ( ) = ( ) ( ) (22) we have our statistical operator ρ for the physical system i of interest and the purification of (19) with (20) is just a The latter equation, really a special case of (21), is known mathematical procedure that might be helpful in perform- as von Neumann’s equation of motion, the quantum analog ing a certain calculation but has no physical significance of Liouville’s equation of motion for phase-space densities. whatsoever. Since the von Neumann equation concerns solely the Yet, the “church of the larger Hilbert space” offers parametric time dependence of ρ, it does not matter to membership to those who believe that there is always such whichcommontimet0 the dynamical operators refer, 7 We are adopting Süßmann’s fitting terminology who distin- ∂ 1 ρ Z(t0),t = H Z(t0),t ,ρ Z(t0),t , (23) guishes between Gemisch (mixture) and Gemenge (blend) [14]. ∂t i Page 6 of 16 and since a unitary transformation relates the dynamical if we collect the probability amplitudes ψ(a, t)inthecol- operators at time t to those at time t0,wealsohave umn Ψ(t), the wave function,andthematrixelementsof H in the matrix H. The corresponding matrix version of pk(t)=tr Πk Z(t),t ρ Z(t),t the von Neumann equation itself is then =tr Πk Z(t0),t ρ Z(t0),t . (24) ∂ 1 According to standard terminology, these two expres- = H, , (33) ∂t i sions for pk(t) are “in the Heisenberg picture” and “in the Schrödinger picture”, respectively. The central von where = ΨΨ† is the matrix that represents the statistical Neumann equation in (22) or (23), however, is picture operator ρ = ||. Of course, (32) is the most familiar independent. The parametric time dependence of the sta- version of the Schrödinger equation, as it applies to a wave tistical operator is what it is. function. The product Πkρ is also an operator that obeys Expressions such as (29) may convey the false impres- Heisenberg’s equation of motion (21), and the two version that the statistical operator depends on time. Yes, sions of (24) give its matrix elements do depend on time, whereas we have d d d ∂ ρ = 0 for the time dependence of ρ.Itisthetime- pk(t)=tr (Πkρ) =tr (Πkρ) (25) dt dt dt ∂t dependent basis of bras that gives time dependence to the wave function in (30), whereas the state ket |is time for the time derivative of the kth probability. The differ- independent. In ence of the two traces is the vanishing trace of the com- mutator of Πkρ with the Hamilton operator. In this sense, |= |a, tψ(a, t)= |a ,tψ(a ,t), (34) then, the Heisenberg picture pays attention to the full time a a dependence, whereas the Schrödinger picture cares only about the parametric time dependence. This is particu- which holds for any time t or t, the basis kets |a, t and larly well visible when the probability operator Πk has no |a,t as well as the probability amplitudes ψ(a, t)and parametric time dependence (the same set-up is used at ψ(a,t) change in time, but |does not. This is, of course, different times), so that (25) becomes ρ || as it should be because = represents our knowl- d dΠk ∂ρ edge about the preparation of the system and that makes pk(t)=tr ρ =tr Πk (26) reference to some definite instant in the past when this dt dt ∂t knowledge was acquired. after the product rule is applied, In passing, we note that some textbook authors have | d dΠk ∂ ∂ρ failed to distinguish clearly between the state ket (Πkρ)= ρ, (Πkρ)=Πk . (27) (an abstract object) and its wave function Ψ (a basis- dt dt ∂t ∂t dependent numerical representation of |), between the The first of these statements is an identity that is generally H H ∂ Hamilton operator and its matrix , and between valid, the second holds only when Πk =0. the statistical operator ρ and its matrix .Intheir ∂t ∂ If we denote by a, t| the common eigenbras of a max- books, then, (32) is written as i |Ψ(t) = H|Ψ(t), A t ∂t imal set ( ) of commuting dynamical variables, a subset as if there were a time-dependent state ket |Ψ(t) that Z t a of the collection ( ), with symbolizing the set of eigen- obeys a Schrödinger equation. As a consequence, it then a, t|A t aa, t| values, ( )= , then we have the Schrödinger seems that we have a time-dependent statistical operator equation ρ |Ψ t Ψ t | ρ ∂ = ( ) ( ) although, in fact, is constant. The re- i a, t| = a, t|H Z(t),t (28) lated failure of distinguishing between the total and the ∂t parametric time dependence then gives rise to the mis- for the relation between a, t +dt| and a, t|.Thematrix leading statement that equations such as (26) demonstrate elements of a rank-one statistical operator, ρ = ||,in that “in the Schrödinger picture ρ depends on time and a this basis, Πk is constant, whereas ρ is constant and Πk depends on a, t|ρ|a,t = ψ(a, t)ψ(a,t)∗, (29) time in the Heisenberg picture” and the like. are products of the probability amplitudes 8 ψ(a, t)=a, t| (30) 5.2 Time-reversal symmetry and irreversible evolution and their complex conjugates. The von Neumann Let us consider the Schrödinger equation for the position equation (22) is then equivalent to wave function of a particle of mass M that moves along ∂ the x axis under the influence of the forces associated with ψ a, t a, t|H Z t ,t |a,tψ a,t V x i ∂t ( )= ( ) ( ) (31) the potential energy ( ), a ∂ or Ω ψ x, t , ∂t +i ( )=0 (35) ∂ i Ψ(t)=HΨ(t), (32) ∂t 8 This section follows [17] closely. Page 7 of 16 where the differential operator Ω is given by A solution is found when we have ⎛ ⎞ ∂ 2 Ω 1 V x > . T = + ( ) 0 (36) ⎜ i ⎟ 2M i ∂x U T,T † ⎝ tH t ⎠ ( ) =exp + d 2( ) Here, for simplicity, the physical requirement that the en- T < ergy is bounded from below is equivalently replaced by ⎛ ⎞ insisting on the positivity of Ω. The time-reversed wave T ⎜ i ⎟ function =exp˙ ⎝− dtH1(t)⎠ = U(T,0) (43) ∗ (ΘT ψ)(x, t)=ψ(x, 2T − t) (37) 0 > is also a solution of the Schrödinger equation (35), ∂ with standard time ordering in the second line and the Ω ψ x, T − t ∗ . ∂t +i ( 2 ) =0 (38) reverse order in the first, and the dotted equal sign means “same effect on ρ ”, that is This time-reversal symmetry of the Schrödinger equation † † does not imply, however, that quantum evolution is re- U(T,T) ρU(T,T)=U(T,0) ρU(T,0) . (44) versible. One can imagine the undoing of all changes in ψ(x, t) In view of the positivity of both H1(t)andH2(t)this that have occurred between, say, t =0andt = T by first can only be realized under exceptional circumstances. The ρ H1 t applying ΘT , followed by evolution under the same dy- question is then: For which and ( )isitpossibleat namics for Tspin state is recovered? That is, we make no at- Ω and this is unitary. But for most the right-hand side is tempt at undoing in full the changes in the center-of-mass − ΩT not of the form e i with a physical, positive Ω.Itfol- state of the atom. References [18,19] deal with the details lows that the undoing of all evolutionary changes is impos- of such a Stern-Gerlach interferometer; it is found that sible, at least if one just wants to exploit the time-reversal one needs three more SGAs for the beam reunion. Ideally, symmetry of the Schrödinger equation. the four SGAs would be perfectly identical. With due re- A more general scheme, not so tightly bound to ele- spect to rule (45b), however, we have to ask how large a mentary nonrelativistic quantum mechanics, is what one mismatch between the SGAs can be tolerated. could call the Humpty-Dumpty problem, which name refers Two relevant quantities are the net transverse mo- to the rhymed riddle (solution: egg) that succinctly ex- mentum transferred to either partial beam and their net presses folk wisdom about irreversibility9. Here, then, transverse displacement, respectively measured by is the t Δp tF t Δz − tF t , Humpty-Dumpty problem = d ( )and = d ( )M (46) Given a statistical operator ρ and a Hamilton operator H1(t) > 0, acting from t =0 to where F (t) is the force on the up-component, for instance, t = T , find T and H2(t) > 0, acting from t = T produced by the inhomogeneity of the magnetic field. Δp Δz to t = T, such that the unitary evolution Thus, and are properties of the macroscopic SGAs; (42) the deviations from the ideal values Δp =0,Δz =0are operators U(T,0) and U(T,T ), constructed as time-ordered exponentials from H1(t)and 10 More generally, one could also consider ancilla-assisted evo- H2(t), respectively, give an over-all evolution lution between T and T, so that the net effect on ρ would operator U(T, 0) = U(T,T )U(T,0) that has come from a completely positive map that may not be describ- U T,T no effect on ρ: U(T, 0) ρ = ρU(T, 0).10 able by a single unitary operator ( ). The main conclusion that, as a rule, quantum evolution is irreversible is not affected, 9 You can find the riddle in [18]. however. Page 8 of 16 resulting from the lack of control over the SGAs. It turns where Kk is the appropriate Kraus operator [20] for the out that to maintain spin coherence one must, at least, probability operator Πk, ensure that † Πk K K . 2 2 = k k (50) δz Δp/ + δp Δz/ 1 (47) Depending on how the POM is implemented, there can holds, where δz and δp are the natural spreads of the beam be quite different Kraus operators for the probability op- prior to entering the first SGA. Therefore, one can toler- erator of interest. In the example of Section 3, we have ate only such variations in the magnetic field that both Δz /δp and Δp /δz, with the consequence K1 = |vac η1/2 H| (51) Δz Δp for detector D1 and similar expressions for D2,D3,and . D4. Irrespective of which detector clicks, we get δz δp (48) ρ →|vacvac| (52) This states that the macroscopic magnetic field must be controlled with sub-microscopic precision – an impossible upon reducing the state, thereby correctly stating that feat. The spin state is so severely affected by the inho- the photon has been absorbed and we are left with the “no mogeneous magnetic field of the first SGA that it shares more photon” situation of the photon vacuum, symbolized Humpty Dumpty’s fate: One cannot undo the damage. by the state ket |vac. More complicated cases are possible, such as a sum of several K†K terms instead of the single term in (50). 6 State reduction The folklore that “a measurement leaves the system in the relevant eigenstate of the observable” applies only to Our statistical operator ρ represents our knowledge about over-idealized projective measurements (meaning that the K the preparation of the system, and this knowledge does ks are pairwise orthogonal projectors). It is puzzling that not change until we acquire additional information. When some textbook authors consider it good pedagogy to ele- that happens, we must update the statistical operator to vate this folklore to an “axiom” of quantum theory. account for the change in the conditions under which we “Collapse of the state” or “wave function collapse” are predict the results of future experiments. State reduction popular synonyms for state reduction. The connotation is the technical term for this updating. that the transition in (49) is a dramatic dynamical pro- State reduction is not particular to quantum theory, it cess, as if the physical system were evolving, is clearly mis- is part and parcel of all statistical formalisms. Bookmakers leading. The various mathematical models that, exactly know about state reduction, even if they never heard the or approximately, accomplish the state reduction (49) as term and have not studied quantum theory. a process in time result from a category mistake: The sta- Here is an example from the world of chess, show- tistical operator is not a physical object or a property of a ing two cases of state reduction11: After eleven rounds of physical object, it describes the object by encoding what the 2013 candidates tournament in London that would we know about it, and state reduction is the bookkeeping determine the challenger of world champion Anand, a device for updating the description. An electron carries its prediction based on past results among the participants, mass and charge around, but not “its” statistical operator. their ratings, and perhaps other data gave a chance It is worth recalling here that different physicists may very of 87% to Carlsen to win the competition, a chance of 7% well use different, equally correct, statistical operators for to Kramnik, and a chance of 6% to Aronian. In the predictions about that same electron. twelfth round, Carlsen lost against Ivanchuk and Kramnik won against Aronian, which resulted in updated chances of 65% for Kramnik and 35% for Carlsen, with Aronian 7 No action at a distance out of the race. Then, in the thirteenth round, Carlsen won against Radjabov and Kramnik drew his game with Here is a quote from a recent news item: Gelfand, and the probabilities for winning the tournament “Quantum theory makes the distinctive pre- became 84% for Carlsen and 16% for Kramnik. The highly diction that non-local correlations are instant: improbable happened in the final fourteenth round: Both for example, a measurement of the polariza- Carlsen and Kramnik lost against Svidler and Ivanchuk, tion of one of a pair of quantum-entangled pho- (53) respectively, and Carlsen took first place. tons should immediately set the polarization k After the th detector of a POM clicked, we update of the other, no matter how far the photons the statistical operator in accordance with are apart, without either photon’s polarization being in any way predetermined.” K ρK† ρ → k k , (49) pk In fact, quantum theory makes no such prediction. There is no instant action at a distance. 11 This is a slightly simplified account; for more details, see The situation alluded to is of the kind depicted in Fig- the news items on www.chessbase.com, 29–31 March 2013. ure 3. A source of entangled photon pairs (SEPP) emits Page 9 of 16

...... D1 ...... D1 ...... D2 ...... Alice’s . . . . Bob’s ...... D2 ...... 12...... SEPP ...... D3 ...... apparatus. . apparatus...... D3 ...... D4 ...... D4 Fig. 3. A source of entangled photon pairs (SEPP) sends photon 1 of each pair to Alice’s laboratory and photon 2 to Bob’s laboratory. Alice’s apparatus detects photon 1 before photon 2 reaches Bob’s apparatus. correlated photons in pairs, with photon 1 sent to Alice different knowledge they have about the situation. Alice and photon 2 to Bob who, for the sake of the argument, is knows the outcome of the measurement on photon 1, Bob assumed to be much farther away from the SEPP than she does not. And we must not fall into the trap of concluding is. For the purpose of this discussion, we allow ourselves that Alice’s statistical operator is better than Bob’s; both (A) (B) the idealization that the photon pairs are described by a her ρ2 and his ρ2 give correct statistical predictions, as rank-one statistical operator, specifically can be verified by many repetitions of the experiment12. So, what about the quote (53)? It is another instance ρ |eprbeprb| 1&2 = (54) of the category mistake noted in Section 6. A change in Alice’s description of photon 2 is misunderstood as a phys- with an Einstein-Podolsky-Rosen [10] state ket in Bohm’s ical process affecting photon 2, and then ex falso sequitur version [21], quodlibet, namely the false assertion about instantaneous 1 1 nonlocal correlations, exemplifiedbytheimmediateset- |eprb = √ |HV−|VH = √ |RL−|LR , (55) ting of the polarization of a photon that can be arbitrarily 2 2 far away. where, for instance, the ket |HV stands for “photon 1 is H- polarized, photon 2 is V-polarized”. Alice and Bob measure their respective photons by four-outcome POMs of the 8 No nonlocality construction shown in Figure 1, with her detector clicking much earlier than his. Nonlocal correlations are a familiar consequence of local Bob’s statistical operator for photon 2 is that of a com- actions: A timing signal from a radio station synchronizes pletely mixed polarization state, clocks at large distances from each other. These nonlocal correlations are clearly not instant; they have a common (B) 1 1 local cause. But then there are claims that quantum sys- ρ2 =tr1{ρ1&2} = |HH| + |VV| = |RR| + |LL| . 2 2 tems can exhibit correlations of nonlocal origin. (56) Recall, for instance, the abstract of a recent paper, This is also Alice’s statistical operator for photon 2 prior which begins with these words: to measuring photon 1. Now suppose Alice’s detector D1 clicks for photon 1. “Bell’s 1964 theorem, which states that the Then, her reduced state for the photon pair is: predictions of quantum theory cannot be ac-

(A) counted for by any local theory, represents (60) ρ = |vac Vvac V|, (57) one of the most profound developments in the foundations of physics.” meaning that really only photon 2 is left, and Alice’s updated statistical operator for that photon is: We submit: Doesn’t quantum theory itself, which is a local theory, account for its own predictions? (A) (A) ρ2 =tr1 ρ = |VV|. (58) As the authors of this quote know very well, experimental data contradict Bell’s theorem [23,24]13,which implies that – as a statement about physical systems – Has anything happened to photon 2 when Alice detected 14 photon 1 with her apparatus? Has “the measurement. . . the theorem is wrong . Since there is no error in the rea- immediately set the polarization” of photon 2? Of course soning that establishes the theorem from its assumptions, not. The transition the flaw must be in the assumptions. Specifically, it is the assumption that a mechanism exists that determines 1 which detector will click for the next photon registered |HH| + |VV| →|VV| (59) 2 by an apparatus of the kind depicted in Figure 1. There is no such deterministic mechanism – quantum processes is Alice’s bookkeeping, it is not a physical process that photon 2 is undergoing, and it happens when she updates 12 A more complicated example is discussed in Section 4.3 her statistical operator. of [22]. Incidentally, we have here an example of the situation 13 Reference [23] reprinted in [11]. mentioned twice above, at the end of Section 3.3 and at 14 We leave it as a moot point whether the theorem is “one the end of Section 6: Alice and Bob use different statistical of the most profound developments in the foundations of operators for predictions about photon 2, reflecting the physics”. Page 10 of 16 are fundamentally probabilistic, events are randomly re- theories: All interactions come about by local couplings; a alized – and the violation of Bell’s theorem by actual data continuity equation states that probability is locally con- confirms that. served; energy, momentum, angular momentum, and other Insights of this kind predate Bell’s work by decades. properties are transferred between particles by highly lo- Recall, for example, the matter-of-fact statement in calized scattering events, and so forth. And just like the Schrödinger’s essay of 1935 [25–27], which reads other local theories do, quantum theory predicts nonlocal correlations that originate in local processes. “[If I wish to ascribe] to all determining parts It is true that joint probabilities that result from quan- definite (merely not exactly known to me) nu- tum processes can have stronger correlations than those merical values, then there is no supposition as (61) available by nonquantum simulations; a violation of Bell’s to these numerical values to be imagined that theorem or one of its variants tells us when such a situa- would not conflict with some portion of quan- tion is at hand. This can be exploited for various purposes, tum theoretical assertions.” including quantum key distribution [29], and experimental 15 realizations [30] have no nonlocal elements. Accordingly, in Trimmer’s translation [28] . there are real-life practical applications of Bell’s theorem, In a derivation of Bell’s theorem, or any of the many and they give Bell’s work lasting value. variants on record, the formalism of quantum theory is We close this subject with noting that a very ba- not used. Instead, one relies on common-sense arguments sic notion of Local Quantum Physics (the title of Haag’s of a certain plausibility in an attempt at describing quan- book [31]) is precisely the locality concept of Bell’s theo- titatively the properties of the joint probabilities in exper- rem, namely that “the free-will decisions by Alice should iments similar to that of Figure 3. Part of that common have no influence on the click frequencies that Bob records sense is a certain notion of locality: If the beam splitter in in his experiment, and vice versa”. In the technical terms Figure 1 is replaced by a mirror that the experimenter may of relativistic quantum field theory, this is the require- or may not put in place, the free-will decisions by Alice ment that the local observables in Alice’s space-time re- should have no influence on the click frequencies that Bob gion commute with the local observables in Bob’s region, records in his experiment, and vice versa. which is at a space-like separation from Alice’s. Since the The actual joint probabilities – which are experimen- notion of locality that enters Bell’s theorem is exactly the tally determined by measuring many pairs emitted by the notion of locality in quantum field theory, the observed SEPP and are correctly accounted for by quantum the- violation of Bell’s theorem by quantum systems cannot ory – do not obey the restrictions that follow from that support the claim (62). common-sensible adhockery, and why should they? Find- ings of an inadequate nonquantum formalism are irrelevant for quantum physics. If the findings are at variance 9 Heisenberg’s cut with the experimental data, as is the case here, we are reminded of the inappropriateness of the reasoning. It fol- 9.1 The cut is where you put it lows that common sense of that sort does not apply in the quantum realm. Alice and Bob have conducted the experiment of Figure 3 Rather disturbingly, though, it has become acceptable on a certain number of entangled photon pairs, indepen- to turn the argument into its opposite. It is taken for dently and identically prepared by the SEPP, and now granted that quantum physics should obey such common they report their measurement results to Charlie. What sense, but then that inadequate nonquantum formalism they communicate to him is the sequence of detector clicks needs nonlocal features – or so it seems16. The conclusion observed and the times at which the clicks occurred. These that data enable Charlie to pair Alice’s clicks with Bob’s and so “Quantum theory is nonlocal.” (62) verify that the click statistics observed is consistent with the predictions obtained from applying Born’s rule to this appears inevitable: So reads the opening line of a paper situation. that reports a violation of Bell’s theorem by experimental Alice and Bob will surely not trouble Charlie with ac- data. counts of their minds, their retinas (or eardrums), or other In fact, however, there is no nonlocality in an exper- irrelevant information. Nor will Charlie bother with such iment such as the one sketched in Figure 3 or any other matters when computing probabilities. Yet, exactly this is quantum-physics experiment on record. The SEPP uses what certain dogmatists insist must be done in principle, well-localized short-range interactions for the creation of although in practice they themselves never do it17. the photon pairs and the same is true for the processes by which the photon absorption in one of the detectors 17 Statements like “In principle, I could solve the Schrödinger gives rise to the observed click. Quantum theory is a lo- equation to predict the next solar eclipse.” are empty unless cal theory in exactly the same sense as all other physical you can do it in practice. Something can be impossible in principle, such as drawing a triangular circle or building a so- 15 Reprinted in [11]. called Popescu-Rohrlich box [32], or possible in practice.But 16 One could also conclude that there is no determinis- to declare that something is possible in principle can at best tic mechanism, were this conclusion not prevented by that mean that one is not aware of any reason that would make it common sense. impossible in principle. Page 11 of 16

The BS, the QWP, the PBSs of the apparatus in Fig- on the microscopic side when the Singapore experiment is ure 1 are made of atoms with which the incoming photon evaluated. interacts. The detectors D1, ..., D4 are made of atoms, In his posthumous eloquent polemic Against ‘measure- too, and so is the experimenter’s retina and her brain. If ment’ [33]19 and also on earlier occasions, Bell20 talks we apply quantum theory to the photon, we should also about Heisenberg’s cut as if it were a universal border be- apply it to the atoms of the optical elements, the detec- tween quantum physics and classical physics. Having made tors, ..., the experimenter’s brain. And whystop here? this mistake21, he then concludes that quantum theory is There are also the atoms of your (the reader’s of these ill-defined because it does not specify the location of the lines) retina, optical nerve, and brain. A truly complete cut with the mathematical precision that he demands it account of the experiment should include all that – as a should have. matter of principle, that is, not of practice. Bell is missing three essential points. First, there is No, it shouldn’t. The experiment of Figure 3 inves- a Heisenberg’s cut whenever the quantum-theoretical for- tigates the polarization properties of entangled photon malism is applied in a particular situation: There is no pairs, and the phenomena of interest do not depend on such universal border. Second, the cut is imprecise by its which material is used to realize the BS and the other nature, it can be shifted to include as many quantum de- components of the four-outcome POM of Figure 1. Yes, if grees of freedom as one can handle in a calculation or one wishes to do so, one can extend the quantum descrip- control in an experiment: Heisenberg’s cut is where you tion beyond the crucial degrees of freedom of the photons, put it. Third, the distinction between microscopic phe- but nothing relevant will be added to Charlie’s analysis of nomena and macroscopic phenomena is meaningful even the data. if one cannot give rigorous criteria for the micro/macro In practice, the formalism of quantum theory is always distinction. applied to the relevant quantum variables (almost always We illustrate this remark by the following analogy. The these are few) and the inclusion of irrelevant ones increases concepts of land and sea are meaningful although you can- the effort only but not the benefit18. The identification of not draw, in a unique and natural way, a mathematical line the relevant variables is usually unproblematic and, if in on the beach that has the sea on one side and the land on doubt, one can take more quantum degrees of freedom into the other. You can draw lines such that all of the sea is account than might be necessary. There are always plenty assuredly on one side, or all of the land is on the other, of technical details that are important for the success of but there is no line that accomplishes both. the experiment without entering the quantum-theoretical A fourth point is this: Alice and Bob rely on the macro- description of the physical system that is studied. For ex- scopic world around them, with its well-defined properties, ample, the experimenter has to calibrate the detectors and when they record their data and communicate them to determine their efficiencies, perhaps exploiting a mathe- Charlie. One cannot even speak about quantum phenom- matical model of some crucial components, but in the end ena without reference to the rather robust classical-physics the only significant number is the efficiency ηk for the kth environment, in which all human activity happens. Deny- detector. When Charlie analyzes the data, he must be able ing that there is this robust environment, is denying the to trust the efficiencies reported to him, but what kind of obvious. detector is actually used does not matter to him. In summary, then, Heisenberg’s cut is needed and, by The imagined demarcation line between the degrees its nature, it is imprecise, and this is just fine. of freedom that are given a full quantum-theoretical description and the “rest of the universe” (to use a pompous expression) is known as Heisenberg’s cut, and the inclusion 9.2 Decoherence of a few more variables into the quantum-theoretical description than really necessary is the shift of Heisenberg’s Whenever we apply the formalism of quantum theory to cut. a particular physical situation, we identify the relevant In a manner of speaking, one can regard Heisenberg’s degrees of freedom and ignore all the rest. In other words, cut as separating the microscopic realm of quantum we put Heisenberg’s cut where we find it appropriate. This physics from the macroscopic world of classical physics, involves a physical approximation because we disregard but one should not be carried away by the connotations all dynamical effects that arise from interactions between that the micro/macro distinction may bring about. We the quantum degrees of freedom that are treated in full are always referring to a particular context, such as Al- and those on the other side of the cut, often referred to ice and Bob taking data and Charlie analyzing them, as the environment. As a consequence, the predictions we not to a generally valid demarcation line. When apply- make about future measurements on the system are not ing the quantum-theoretical formalism to an experiment in Munich, the polarization degrees of freedom of pho- 19 The replies by van Kampen [34,35], Peierls [36], and ton pairs emitted by a SEPP in Singapore are on the Gottfried [37] must not be missed. 20 macroscopic side of Heisenberg’s cut, whereas they are This one exception from the no-finger-pointing policy of footnote 1 is unavoidable. 18 This is also true for any application of Newton’s classical 21 Bell also makes that category mistake of regarding the wave mechanics, Maxwell’s electromagnetism, or any other physical function as a physical object, and he misunderstands state theory, without the dogmatists crying foul. reduction as a physical process. Page 12 of 16 absolutely reliable and, since small errors accumulate, the link between the formalism and the phenomena, is not predictions are less reliable for the far future than the near what they are questioning. They would refer to that as the future. “minimal interpretation” and that is not enough for them. In terms of the parametric time dependence of the sta- They want to know “what is really going on”, thereby re- tistical operator, this means that the von Neumann equa- quiring answers to questions that quantum theory cannot tion (23) applies only during the initial period when the provide. residual interactions with the environment can be ignored. These are questions about the mind-body problem,the If we wish to extrapolate from the initial time t0 to much relation between our conscious human experiences and our later times, the uncertainty that originates in our lack free will on one side, and the reality of natural phenom- of knowledge about the system-environment interactions ena on the other – philosophical issues that are outside of across Heisenberg’s cut has to be taken into account. This physics. Yes, it is important to understand the philosoph- leads to a degradation of our statistical operator at later ical implications of the lessons of quantum physics, but times t, usually noticeable as an increase of the entropy. one must not confuse physics with philosophy. There is a rich literature about this so-called decoher- The “lack of consensus” noted in (63) is not about ence, and many ways of modeling the environment and the interpretation of quantum theory in the strict sense of the system-environment interactions have been proposed Section 1, but about its bearing on philosophy. The use and studied (a simple example of “phase decoherence” ap- of the term “interpretation” for the more or less success- pears in Sect. 11.1). When a suitable model is adopted, the ful reconciliation of philosophical preconceptions with the resulting modification of the von Neumann equation (23) quantum-physical phenomena is most unfortunate, and can give quite an accurate account of the loss in preci- there is little hope that this practice will change. Fuchs’s sion of our predictions about the future properties of the and Peres’s verdict Quantum Theory Needs No ‘Interpre- quantum system. tation’ (the title of their essay [42]) should be understood in this context: Quantum theory has its interpretation and does not need a philosophical Uberbau¨ . 10 No murky interpretation Of course, philosophers should be encouraged to study the lessons of quantum theory – for example, the lesson The introduction of a recent paper contains the following that the future is not predetermined by the past24 – but sentences: their philosophical debates are really irrelevant for quantum theory as a physical theory, although they are likely “[. . . ] a century after the discovery of quan- to be of great interest to physicists. We could leave it tum mechanics, it seems that we are no closer at that, were there not the widespread habit of the de- to a consensus about its interpretation than baters to endow the mathematical symbols of the formal- we were in the beginning. The collapse of the ism with more meaning than they have. In particular, quantum state occurring during the process there is a shared desire to regard the Schrödinger wave ofmeasurement[...]doesnothaveanunam- (63) function as a physical object itself after forgetting, or re- biguous definition and a reasonable explana- fusing to accept, that it is merely a mathematical tool that tion. [. . . ] some radical changes in our classi- we use for a description of the physical object (electron, cal understanding of reality have to be made; atom, photon, ...). The regrettable textbook practice e.g. constructing a physical process of collapse, of invoking analogies between Schrödinger’s probability- accepting the existence of parallel worlds, or amplitude waves and sound waves, water-surface waves, adding nonlocal hidden variables.” or electromagnetic waves provokes and feeds that desire. Of course, the authors of these lines use Born’s rule for The fundamental difference is then ignored: Sound waves, the calculation of probabilities, associate the wavelengths water waves, electromagnetic waves are physical objects of spectral lines with differences of energy eigenvalues, and with mechanical properties, they carry energy, momentum, angular momentum from here to there, whereas the use all other standard links between the mathematical for- 25 malism and the physical phenomena. In other words, these Schrödinger waves are mathematical symbols . authors’ interpretation of quantum theory is the usual The reification of the wave function requires that the one22. Why, then, is there this blunt assertion that “there “collapse of the quantum state” is a physical process, is no consensus about the interpretation”? but since it isn’t, there cannot be “an unambiguous def- Clearly, the authors of (63) have a nonstandard mean- inition”, and a “reasonable explanation” is not possi- ing of the word “interpretation” in mind, and so have oth- ble. We submit: With such incorrect meaning given to ers23. The actual interpretation, that is: The battle-tested gave about that many “interpretations” to choose from, while 22 It is simply the interpretation of quantum theory. Calling it Tegmark’s poll of 1998 [41] had only half as many choices. the standard interpretation or using any other identifier is un- 24 However the next photon is prepared, there is no way of necessary and, indeed, harmful because “this sounds as if there knowing which of the four detectors in the experiment of Fig- were several interpretations of quantum mechanics. There is ure 1 will click. only one”. (Peierls as quoted by Davies and Brown [38].) 25 Also, only very special cases of wave functions are 23 The Compendium of Quantum Physics [1] has entries on probability-amplitude waves in the three-dimensional space of more than ten “interpretations”, and two recent polls [39,40] physical positions. Page 13 of 16 the wave function and to the process of state reduc- definite click of one of the detectors and why√ do we never tion, one no longer deals with quantum theory, but with see superpositions such as |C1 + |C2 / 2withthetwo something else. The conclusions that are then reached detectors having half-clicked? about philosophical implications are of no consequence. Although these questions seem to be addressing pro- Indeed, the alleged implications (“parallel worlds”) are found issues, they are not formulating a serious “mea- bewildering and prompt us to paraphrase van Kampen’s surement problem”. At best, they are rephrasing matters Theorem IV [43]: If you endow the mathematical symbols already discussed. with more meaning than they have, you yourself are re- First, we note that the state ket |in (68) is the same sponsible for the consequences, and you must not blame as the one in (66). The two versions of |are particular quantum theory when you get into dire straits. examples of the general statement in (34). Both versions The interpretation of quantum theory is not murky, it of |encode what we know about the physical situation. is absolutely clear; there is no lack of consensus about it. It Why, then, are all those questions asked about (68), but is one of the three defining constituents of quantum theory none about (66)? √ (Sect. 1), and all practicing quantum physicists rely on it Second, the superposition |C1 + |C2 / 2doesnot all the time. describe a situation where detectors D1 and D2 each have “half-clicked” (whatever that would mean) but an entirely new situation that is√ outside our experience. Re- 11 You cannot see what you member that |V + |Hi / 2 is not a polarization state wouldn’t recognize that is “half-vertical” and “half-horizontal” but one that is entirely right-circular, phenomenologically very differ-√ ϕ 11.1 No measurement problem ent from both |V and |H. We know what |V+|Hei / 2 means (as a statement about a photon’s polarization) for We can use the set-up of Figure 1, see Sections 3.1 and 3.2, any value of the √ phase ϕ, but we have no clue about iϕ for a discussion of the so-called “measurement problem”. |C1+|C2e / 2. So, we never see these superpositions A typical argument would develop along the following because we cannot recognize them26.Ifwecould,either lines. At the initial time tini, we have the photon approach- version of |would enable us to calculate the probability ing the BS with a polarization-state ket of seeing these superpositions. 2 2 Third, despite strong assertions that it does, decoher- |℘ = |Vα + |Hβ with α + β =1, (64) ence does not come to the rescue. The asserters consider the statistical operator associated with ket |D, and the four detectors are in the “ready” state

ρD = |DD| (70) |R = |R1R2R3R4, (65) 4 so that we write = |NCp0NC| + |CkpkCk| + cross terms , k=1 |= |℘ R,tini (66) where the cross terms collect all ket-bra products of two for the state ket as characterized by properties referring different outcomes, to the initial time. Expressed with reference to the final 4 time tfin, this state ket is √ √ cross terms = |NC p0pkCk| + |Ck p0pkNC| 4 k=1 √ √ √ |= |vac NC,tfin p0 + |vac Ck,tfin pk, (67) + |Cj pjpkCk|. (71) k=1 j=k where Ck stands for “the kth detector has clicked and the | other three have not” and NC denotes the no-click situa- At this point, one remembers that Ck symbolizes a state tion. At the final time, the photon has been absorbed and of very many quantum degrees of freedom (all those nuclei we have the photon vacuum state for all detector states, and electrons that make up the optical elements and the which allows us to focus on the final state of the detectors, detectors) and, therefore, one is suddenly unsure about details such as the over-all phase assigned to |Ck,andso it appears natural to refine the quantum-theoretical de- |= |vac D,tfin (68) iφk scription by the replacements |Ck→|Cke with ran- with dom phases φk. Here, “random” simply means that the √ 4 √ |D |NC p0 |Ck pk. 26 = + (69) He who interjects “But, in principle, I could certainly k=1 measure the observables described by the hermitian operators iϕ −iϕ This is the point where some start to worry: The detec- |C1e C2| + |C2e C1|, and when the eigenvalue +1 is tors are in a superposition of this one having clicked or found, I will have succeeded.” should reread footnote 17 and that one having clicked, then how does one account for a then ask himself Can I do it in practice? Page 14 of 16 phases acquire different unknown values whenever the ex- probabilities by looking at the weights of our preferred as- periment is repeated. Averaging over the random phases if reality. Such a reading-off can sometimes give the right removesthecrossterms, answer, but we know that it does only after an application of Born’s rule has confirmed our educated guess. cross terms → 0, (72) Fourth, the transition (73) is an improvement of our description, it is not a physical process. The decoherence and we arrive at an effective statistical operator for the iφk detectors, argument with the unknown phase factors e and so forth is a confession that we don’t know the dynamics with 4 sufficient precision, and our starting point (66) is wrong, ρD → ρD, eff = |NCp0NC| + |CkpkCk|. (73) too. We do not have enough information for describing k=1 the initial “ready” state of the detectors by a state ket |R or its corresponding rank-one statistical operator |RR|. The adjective “effective” means that all observable prop- We can, at most, have limited knowledge about a few de- ρ erties calculated with D,eff are the same as those calcu- grees of freedom of the detectors and account for that by lated with ρD. In other words, the cross terms of (70) are a highly mixed state. The state kets of (66), (68), and (69) ineffective, they are of no consequence. This, of course, border on utter nonsense. is not a recent insight. There are, for example, the ac- 27 When preparing the experiment and getting the de- counts by Süssmann [14] and Peres [44] of 1958 and 1980, tectors into the “ready” state, we are not controlling all respectively. quantum degrees of freedom of the detectors perfectly but, ρ ρ So, D “has decohered” into D,eff – the transition (73) at best, the few relevant ones with some limited precision. is now regarded as a physical process! – and we have a There are, therefore, very many pure quantum states that mixed state that is as if we had a Gibbs ensemble com- we could imagine for the “ready” detectors rather than a posed of a fraction p0 of no-click cases, a fraction p1 of unique state associated with the ket |R. Strictly speak- clicks by detector D1,...,afractionp4 of clicks by de- ing, there is not even the pure-polarization ket |℘, but tector D4. One of the situations is the case, and we do that is acceptable as a pretty good approximation in a not know which one until we take note of the outcome of well-controlled experiment. the experiment. All is fine, or so it seems, because we just Rather than the convex sum of projectors in (73) and have this “classical ignorance” rather than the quantum- (74), we have, at best, a convex sum of highly mixed sta- ρ theoretical indefiniteness of D, and we no longer need to tistical operators for the detectors, worry why we do not see the detectors with properties √ described by superpositions such as |C1 + |C2 / 2. 4 (NC) (k) All is fine? No, we are fooling ourselves. Remembering ρD = p0ρD + pkρD , (76) Section 4.1, we note that the effective statistical operator k=1 does not have this one unique as-if reality associated with (k) (NC) it, but very many of them. For example, we have where ρD is for a click of the kth detector and ρD for the no-click case. But this ρ , too, is more an illusion ρ | p | | p | | p | D D,eff = NC 0 NC + C1 1 C1 + C2 2 C2 than a reality. Even the smallest detectors contain a gar- p3 + p4 gantuan number of atoms, and we don’t really know what + |C34+ C34+| 2 exactly the detectors are composed of – removing a single p3 + p4 molecule, say, from the surface of one of the wires in the + |C34− C34−| (74) 2 apparatus has no bearing on the relevant aspects of the physical situation. Therefore, we will be hard-pressed to with ρ(k) 28 write down meaningful expressions for the D s. p3 p4 This is not a problem, though, because there is no |C34± = |C3 ±|C4 (75) p3 + p4 p3 + p4 use of the quantum description of the detectors that have clicked or not. There are no probabilities of subsequent and, clearly, the mixture described by ρD,eff can also be events to be calculated from ρD. The photon-absorption blended by p0 parts of no-click cases, p1 parts of clicks event has been amplified or not, one of the detectors has 1 by D1, p2 parts of clicks by D2,plus 2 (p3 + p4)parts clicked or the photon has escaped detection: The measure- each for the situations (of unknown phenomenology) that ment of this photon’s polarization is over. are described by the superpositions |C34+ and |C34−.As Fifth, since neither decoherence nor any other mecha- always, there is an infinity of blends for the one mixture nism select one particular outcome (see Sect. 8), the whole and a corresponding infinity of as-if realities. “measurement problem” reduces to the question Why is Another reason why we are fooling ourselves is that we there one specific outcome? which is asking Why are there still need the principle of random realization for selecting randomly realized events? in the particular context con- one of the five possibilities and Born’s rule for calculating sidered. This harkens back to Section 1, where we noted the probabilities from ρD,eff . It is not the case, as self- that quantum theory cannot give an answer. suggesting this may appear, that we can just read off the 28 He who is still pondering the question in footnote 26, may 27 Reprinted in [11]. find a useful hint here. Page 15 of 16

In summary, then, the alleged “measurement problem” wouldbecalledaSchrödinger-cat state in today’s jargon. does not exist as a problem of quantum theory. Those who But that is not the case. The term is routinely applied want to pursue the question Why are there events? must to superpositions of states of a few quantum degrees of seek the answer elsewhere. freedom, or even of a single one, that have vastly different quantum numbers. In this sense, the silver atom emerging from a Stern-Gerlach apparatus is in a Schrödinger-cat 11.2 Schrödinger’s cat state. This usage of the term does not do justice to what Schrödinger had in mind; it should be discouraged, the In this context one can hardly avoid the mentioning more so because such superpositions are nothing special, 29 of Schrödinger’s infamous cat . When commenting on they are common in standard interferometric devices. the physical significance of the quantum-theoretical wave function – it is an expectation catalog (Katalog der Erwartung)inSchrödinger’s own words – he gave, as a 12 Summary warning, the “ridiculous” (burlesk) example of a cat whose vital state serves as the indicator whether a radioactive The foundations of quantum theory have been laid, and decay has occurred or not. That passage reads laid well, by the founding fathers – Planck, Einstein, Bohr, Heisenberg, Schrödinger, Born, Dirac, and others. Much “One can even set up quite ridiculous cases. A has happened since those early days of the 1920s. The cat is penned up in a steel chamber, along with phenomenology and the formalism are much richer today the following diabolical device (which must be and, to supplement this enrichment, the interpretation has secured against direct interference by the cat): more links between the formalism and the phenomena. in a Geiger counter there is a tiny bit of ra- This is a natural consequence of our communal duty to dioactive substance, so small, that perhaps in explore the consequences, implications, and applications the course of one hour one of the atoms decays, of quantum theory, and we are still in the process of doing but also, with equal probability, perhaps none; just that. if it happens, the counter tube discharges and It is true that, occasionally, the founding fathers and through a relay releases a hammer which shat- (77) their peers found it difficult to reconcile the lessons of ters a small flask of hydrocyanic acid. If one quantum theory with the world view they had acquired has left this entire system to itself for an hour, earlier. The younger players, notably Heisenberg and one would say that the cat still lives if mean- Dirac, overcame these difficulties more quickly and more while no atom has decayed. The first atomic easily than their seniors, Planck and Einstein among them. decay would have poisoned it. The ψ-function Einstein, for example, did not like the random nature of of the entire system would express this by hav- events. He told Franck the following30: ing in it the living and the dead cat (pardon the expression) mixed or smeared out in equal “I can, if the worse comes to worst, still realize parts.” that God may have created a world in which there are no natural laws. In short, chaos. But in Trimmer’s translation [25–28]. In other words, the dia- that there should be statistical laws with def- (79) bolic device establishes the strong correlations inite solutions, i.e., laws that compel God to throw dice in each individual case, I find highly no decay ←→ live cat, disagreeable.” yes decay ←→ dead cat, (78) While Einstein’s metaphor of the dice-throwing God is so that the cat is a bona fide detector for the radioac- charming and memorable, and his personal difficulties are tive decay event. It has the obvious advantage that the of great historical interest, it is ultimately irrelevant what detector is easy to read. he, or any other individual, finds disagreeable. The facts Schrödinger is, of course, overshooting the mark. Just count – and Einstein, of course, understood this. In sci- like there is no ket |D for the detectors in Section 11.1, ence, truth is the daughter of time, not of authority. there is no ψ-function that has “the living and dead cat One must also acknowledge that, after the lapse of smeared out”. The imagined superpositions of the dead fourscore years, the terminology is more precise today and the live cat, which have inspired so many authors, are than it was during the founding years. Concepts get clearer in time, and one learns to avoid sloppy terms and mis- fantastic phantoms of the same kind as the √ detector states iϕ leading phrases. The foundations, however, have not been associated with the kets |C1 + |C2e / 2. We do not know how to recognize them and, therefore, we cannot see touched by these refinements. them. In all of that, the cat is really superfluous, it just As way of summary, here are our answers to the ques- adds drama to the plot; the position of the hammer, say, tions asked at the beginning: already tells the story. – Yes, quantum theory well defined. It would be fitting if any such correlated state of a – Yes, quantum theory has a clear quantum degree of freedom and a macroscopic detector interpretation.

29 Heisenberg’s dog [45] does not concern us here. 30 As quoted by Snow in [46]. Page 16 of 16

– Yes, quantum theory is a local theory. 12. N.H.D. Bohr, Nature 136, 65 (1935) – No, quantum evolution is not reversible. 13. N.H.D. Bohr, Phys. Rev. 48, 696 (1935) – No, wave functions do not collapse; you 14. G. Süßmann, Abh. Bayer. Akad. Wiss. 88 (1958) reduce your state. 15. N. Gisin, Helv. Phys. Acta 62, 363 (1989) – No, there is no instant action at a distance. 16. N. Gisin, Phys. Lett. A 143, 1 (1990) – Heisenberg’s cut is where you put it. 17. B.-G. Englert, Z. Naturforsch. 52a, 13 (1997) – No, Schrödinger’s cat is not half dead and 18. J. Schwinger, M.O. Scully, B.-G. Englert, Z. Phys. D 10, half alive. 135 (1988) – No, there is no “measurement problem”. 19. B.-G. Englert, M.O. Scully, J. Schwinger, Found. Phys. 18, 1045 (1988) 20. K. Kraus, States, Effects, and Operations: Fundamental Tersely: Quantum theory is a well-defined local theory Notions of Quantum Theory, volume 190 of Lecture Notes with a clear interpretation. No “measurement problem” or in Physics, edited by A. Böhm, J.D. Dollard, W.K. any other foundational matters are waiting to be settled. Wootters (Springer, Heidelberg, 1983) What, then, about the steady stream of publications 21. D. Bohm, Quantum Theory (Prentice Hall, New York, that offer solutions for alleged fundamental problems, each 1951) of them wrongly identified on the basis of one misunder- 22. A. Buchleitner, K. Hornberger, Coherent Evolution in standing of quantum theory or another? Well, one could Noisy Environments, volume 611 of Lecture Notes in be annoyed by that and join van Kampen [47] in calling Physics (Springer, Heidelberg, 2002) it a scandal when a respectable journal prints yet another 23. J.S. Bell, Physics 1, 195 (1964) such article. No-one, however, is advocating censorship, 24. J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 23, 880 (1969) even of the mildest kind, because the scientific debate can- 25. E. Schrödinger, Naturwissenschaften 23, 807 (1935) not tolerate it. Yet, is it not saddening that so much of 26. E. Schrödinger, Naturwissenschaften 23, 823 (1935) diligent effort is wasted on studying pseudo-problems? 27. E. Schrödinger, Naturwissenschaften 23, 844 (1935) 28. J.D. Trimmer, Proc. Am. Philos. Soc. 124, 323 (1980) The Centre for Quantum Technologies is a Research Centre 29. A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991) of Excellence funded by the Ministry of Education and the 30. A. Ling, M.P. Peloso, I. Marcikic, V. Scarani, A. Lamas- National Research Foundation of Singapore. Linares, C. Kurtsiefer, Phys. Rev. A 78, 020301 (2008) 31. R. Haag, Local Quantum Physics (Springer, Heidelberg, 1996) References 32. S. Popescu, D. Rohrlich, Found. Phys. 24, 379 (1994) 33. J. Bell, Phys. World 3, 33 (1990) 1. Compendium of Quantum Physics: Concepts, Experi- 34. N.G. van Kampen, Phys. World 3, 20 (1990) ments, History and Philosophy, edited by D. Greenberger, 35. N.G. van Kampen, Phys. World 4, 16 (1991) K. Hentschel, F. Weinert (Springer, Heidelberg, 2009) 36. R. Peierls, Phys. World 4, 19 (1991) 2. I. Newton, Philosophiae Naturalis Principia Mathematica 37. K. Gottfried, Phys. World 4, 34 (1991) (Streator, London, 1687) 38.P.C.W.Davies,J.R.Brown,The Ghost in the Atom: A dis- 3. J.C. Maxwell, A Treatise on Electricity and Magnetism cussion of the Mysteries of Quantum Physics (Cambridge (Clarendon Press, Oxford, 1873) University Press, Cambridge, 1986) 4.M.Born,Z.Phys.37, 863 (1926) 39. M. Schlosshauer, J. Kofler, A. Zeilinger, 5. W. Gerlach, O. Stern, Z. Phys. 9, 353 (1922) arXiv:1301.1069[quantph] 6. B. Friedrich, D. Herschbach, Phys. Today 56, 53 (2003) 40. C. Sommer, arXiv:1303.2719[quantph] 7. J. Bernstein, arXiv:1007.2435[physics.hist-ph] 41. M. Tegmark, Fortschr. Phys. 46, 855 (1998) 8. R. Haag, Commun. Math. Phys. 132, 245 (1990) 42. C.A. Fuchs, A. Peres, Phys. Today 53, 70 (2000) 9. R. Haag, Found. Phys. 43, 1295 (2013) 43. N.G. van Kampen, Physica A 153, 97 (1988) 10. A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777 44. A. Peres, Phys. Rev. D 22, 879 (1980) (1935) 45. J.A. Bergou, B.-G. Englert, J. Mod. Opt. 45, 701 (1998) 11. Quantum Theory and Measurement,editedbyJ.A. 46. A.P. French, Einstein: A Centenary Volume (Harvard Wheeler, W.H. Zurek (Princeton University Press, University Press, Cambridge, 1979) Princeton, 1983) 47. N.G. van Kampen, Am. J. Phys. 76, 989 (2008)