Quantum Reversibility Is Relative, Or Do Quantum Measurements Reset Initial Conditions?
QQuantum Reversibility is Relative,OrDo Quantum Measurements Reset Initial Conditions?
Wojciech H. Zurek Theory Division, LANL, MS B213, Los Alamos, NM 87545 (Dated: August 28, 2018)I compare the role of the information in the classical and quantum dynamics by examining the rela-tion between information flows in measurements and the ability of observers to reverse evolutions.I show that in the Newtonian dynamics reversibility is unaffected by the observer’s retention ofthe information about the measurement outcome. By contrast—even though quantum dynamics isunitary, hence, reversible—reversing quantum evolution that led to a measurement becomes in prin-ciple impossible for an observer who keeps the record of its outcome. Thus, quantum irreversibilitycan result from the information gain rather than just its loss—rather than just an increase of the(von Neumann) entropy. Recording of the outcome of the measurement resets, in effect, initialconditions within the observer’s (branch of) the Universe. Nevertheless, I also show that observer’sfriend—an agent who knows what measurement was successfully carried out and can confirm thatthe observer knows the outcome but resists his curiosity and does not find out the result—can, inprinciple, undo the measurement. This relativity of quantum reversibility sheds new light on theorigin of the arrow of time and elucidates the role of information in classical and quantum physics.Quantum discord appears as a natural measure of the extent to which dissemination of informationabout the outcome affects the ability to reverse the measurement.
I. INTRODUCTION
Quantum as well as classical equations of motion arereversible. Yet, irreversibility we, observers, perceive isan undeniable “fact of life”. In particular, quantum mea-surements are famously regarded as irreversible [1]. Thisirreversibility is a reason why modeling of quantum mea-surements using unitary dynamics is sometimes viewedas controversial. Of course, decoherence [2–5] (now usu-ally included as an essential ingredient of a fully con-summated measurement process) is rightly regarded as effectively irreversible. The arrow of time it dictates canbe tied to the dynamical second law [6, 7].Our aim here is to point out that, over and above thefamiliar irreversibility exemplified by decoherence thatstems from the second law, and in contrast to the classicalphysics, irreversibility of an even more fundamental kindarises in quantum physics in course of measurements. Weshall explore it by turning “reversibility” from an ab-stract concept that characterizes equations of motion toan operationally defined property: We shall investigatewhen the evolution of a measured system and a measur-ing apparatus can be, at least in principle, reversed evenif the information gained in course of the measurement ispreserved (e.g., the record imprinted on the state of theapparatus pointer is copied).This operational view of reversibility yields new in-sights: We shall see that reversing quantum measure-ments becomes impossible for an observer who retainsrecord of the measurement outcome. This is becausethe state of the measured quantum system revealed andrecorded by the observer assumes—for that observer—the role reserved for the initial state in the classical, New-tonian physics. Consequently, clear distinction between the initial con-ditions and dynamics—the basis of classical physics [8]—is lost in a quantum setting. Indeed, quantum measure-ments can be reversed only when the record of the out-come is no longer preserved anywhere else in the Uni-verse. By contrast, classical measurement can be re-versed even if the record of the outcome is retained.Irreversibility caused by the acquisition of informationin a quantum measurement has a different origin and adifferent character from irreversibility that follows fromthe second law [6]. There, the arrow of time – the im-possibility of reversal – is tied to the increase of entropy,and, hence, to the loss of information. In quantum mea-surements irreversibility can be a consequence of the ac-quisition (rather than loss) of information.This loss of the ability to reverse is relative —it dependson the information in possession of the agent attemptingreversal. Thus, a friend of the observer, an agent who re-frains from finding out the outcome (but can control thedynamics that led to that measurement) can, at least inprinciple (and in a setup reminiscent of “Wigner’s friend”[9]) undo the evolution that resulted in that measurementeven after he confirms that the observer had—prior toreversal—perfect record of the outcome.Measurements re-set initial conditions relevant for ob-server’s evolution in a manner that is tied to the choice ofwhat is measured (as emphasized by John Wheeler [10],see Fig. 1). Quantum measurements (more generally,“quantum jumps”) undermine one of the foundationalprinciples of the classical, Newtonian dynamics: There,consecutive measurements just narrowed down the bun-dle of the possible past trajectories consistent with ob-server’s knowledge. Thus, in a classical, deterministicUniverse it was always possible to imagine a single ac-tual trajectory that fit within this bundle, and was trace- a r X i v : . [ qu a n t - ph ] A ug FIG. 1. An agent—an observer—within the evolving and ex-panding Universe carries out measurements that help defineinitial conditions of that Universe [10]. Thus, initial condi-tions (at Big Bang) are determined in part by measurementscarried out at present. This dramatic image (due to JohnWheeler) is illustrated by the study of the ability to reversean act of acquisition of information in this paper. able to the point marking the initial condition. Thismeant that evolution was reversible, an that it could beretraced—hence, reversed—using the present state of thesystem as a starting point into the dynamical laws and“running the evolution backwards”.This idealization of a single starting point of “myUniverse”—i.e., the unique Universe consistent with theoutcomes of all the past measurements at observer’sdisposal—is no longer tenable in the quantum setting.Quantum measurement derails evolution, resetting itonto the track consistent with its outcome.The loss of distinction between initial conditions anddynamical laws is tied to the enhanced role of informa-tion in the quantum Universe: Information is not justa passive reflection of the deterministic trajectory dic-tated by the dynamics (as was imagined in the classical,Newtonian settings) but it is acquired in a measurementprocess that changes the state of both the measured ob-ject and of the measuring apparatus (or of an agent /observer).We start in the next section by comparing information-theoretic prerequisites of a successful reversal in thequantum and classical case. In Section III we discussthe use of quantum discord to quantify the inability to reverse measurements. Section IV shows that anotheragent, a friend of the observer, can confirm that the ob-server is in possession of the information about the out-come in a way that does not preclude the reversal anddoes not reveal the outcome. This leads us to concludethat in a quantum world reversibility is indeed relative—it depends on the information in possession of the agent.Discussion and summary are offered in Section V.We note that much of the technical content of the pa-per amounts to the proverbial “beating around the bush”.This is because the key point is “personal” and simple—an agent who is in possession of the information about theoutcome is incapable of undoing the measurement thatled to that outcome. Yet, the tools at our disposal—statevectors, density matrices, unitary evolution operators—constrain us to discuss the measurement process “fromthe outside”. And, from that external vantage point,information retained by the observer or copied into hisrecord-keeping device plays the same role as the informa-tion acquired by the environment in course of decoher-ence or (especially) quantum Darwinism [3, 5, 11]. Onecould even say that we are stuck in the shoes of Wigner’sfriend [9], looking at the observer “from the outside”.The ultimate message of this paper is that the observer/ agent is incapable of undoing the acts of the acquisitionof information, and that this inability to reverse revealsan origin of the arrow of time that is uniquely quan-tum and that is not dependent on the entropy increasemandated by the second law. There is of course no con-tradiction between the resulting arrows of time, and (asdecoherence accompanies quantum measurements [2–6])they generally appear together and point in the same di-rection, but they are nevertheless distinct. One way toexpress this difference is to note that, while our discus-sion is phrased in the language that presumes unitarityof evolutions, this inability to reverse may be easier toexpress using Bohr’s “collapse” imagery [12].
II. RECORDS AND REVERSIBILITY
We study operational reversibility—the ability of anobserver to reverse evolution—in the classical and quan-tum setting. Our goal is to show that, in the quan-tum world, information has physical consequences thatgo far beyond its role in the classical, Newtonian dynam-ics. This illustrates the difference between the nature andfunction of information in quantum and classical physics.The key gedankenexperiment involves a measuredquantum (or classical) system S ( S ), and an agent /apparatus A ( A ). The records from A ( A ) can be fur-ther copied into the memory device D ( D ). We shallnow show that presence of the copy of the record of themeasurement outcomes has no bearing on the (in princi-ple) ability to reverse a classical measurement, but pre-cludes reversal of a quantum measurement. Thus, thepre-measurement state of the classical SA can be re-stored even when D knows the outcome. Such reversalis not possible for a quantum SA as long as D retains acopy of the measurement result.It is important to emphasize the distinction betweenthe usual discussions of reversibility (that focus on thereversibility of the equations that generate the dynam-ics) and our aims: Here we take for granted that it ispossible to implement operators that can undo dynami-cal evolutions (including these leading to measurements)in the absence of any leaks of information. Thus, in asense, we are siding with Loschmidt in his debate withBoltzmann. For instance, we assume observer can switchthe sign of the Hamiltonian that resulted in the measure-ment. Our aim is to shift the focus of attention from thedynamics to the role of the information observer has inimplementing reversals. A. Reversing classical measurement (while keepingrecord of its outcome)
We start by examining measurements carried out bya classical agent / apparatus A on a classical system S .The state s of S (e.g., location of S in phase space) ismeasured (with some accuracy, but we do not need toassume perfection) by a classical A that starts in the“ready to measure” state A : sA E SA = ⇒ sA s (1 a )The question we address is whether the combined stateof SA can be restored to the pre-measurement sA evenwhen the information about the outcome is retainedsomewhere, e.g. copied into the memory device D .The dynamics E SA responsible for the measurement isassumed to be reversible and, in Eq. (1 a ), it is classical.Therefore, classical measurement can be undone simplyby implementing E − SA that is assumed to be at the dis-posal of the observer. And example of E − SA is (Loschmidtinspired) instantaneous reversal of all velocities.Our main point is that the reversal sA s E − SA = ⇒ sA (1 a (cid:48) )can be accomplished even after the measurement out-come is copied onto the memory device D : sA s D E AD = ⇒ sA s D s (2 a )so that the pre-measurement state of S is recorded else-where (here, in D ). Above, E AD plays the same role as E SA in Eq. (1 a ). That is, the examination of S and A separately, or of the combined SA will not reveal anyevidence of irreversibility. After the reversal; sA s D s E − SA = ⇒ sA D s (3 a )the state of SA is identical to the pre-measurement state,even though recording device retains the copy of the out-come. Classical controlled-not gates provide a simple ex-ample of the claims above, as one can readily verify. Starting with a partly known state of the system doesnot change this conclusion. Thus, initial informationtransfer from S to A :( w s s + w r r ) A E SA = ⇒ w s s A s + w r r A r (4 a )when the system is beforehand in a classical mixture oftwo states r , s with the respective probabilities w r , w s can be undone— S and A will return to the initial state—even if an intermediate information transfer from A to D has occurred:( w s s A s + w r r A r ) D E AD = ⇒ w s s A s D s + w r r A r D r . (5 a )This is easily seen: w s s A s D s + w r r A r D r E − SA = ⇒ ( w s s D s + w r r D r ) A . (6 a )In the end S is still correlated with D —that is D has therecord of the outcome of the measurement of S by A .However, anyone who measures the combined state of S and A will confirm that the evolution that resulted inthe measurement of S by A has been reversed. That is,the apparatus / agent A is back in the pre-measurementstate, and the system S has the pre-measurement proba-bility distribution over the classical microstates r,s (evenif they are still correlated with the states of the mem-ory device D ). Thus, in classical dynamics retention ofrecords—presence of information about the outcome ofthe measurement—does not preclude the ability to re-verse evolutions. B. Reversing quantum measurement (can’t keepthe record of the outcome)
Consider now a measurement of a quantum system S by a quantum A : (cid:0)(cid:88) s α s | s (cid:105) (cid:1) | A (cid:105) U SA = ⇒ (cid:88) s α s | s (cid:105)| A s (cid:105) (1 b )The evolution operator U SA is unitary (for example, U SA = (cid:80) s,k | s (cid:105)(cid:104) s || A k + s (cid:105)(cid:104) A k | with orthogonal {| s (cid:105)} , {| A k (cid:105)} would do the job). Therefore, evolution that leadsto a measurement is in principle reversible. Reversal im-plemented by U †SA is possible, and will restore the pre-measurement state of SA : (cid:88) s α s | s (cid:105)| A s (cid:105) U †SA = ⇒ (cid:0)(cid:88) s α s | s (cid:105) (cid:1) | A (cid:105) (1 b (cid:48) )Let us however assume that the outcome of the measure-ment is copied before reversal is attempted: (cid:0)(cid:88) s α s | s (cid:105)| A s (cid:105) (cid:1) | D (cid:105) U AD = ⇒ (cid:88) s α s | s (cid:105)| A s (cid:105)| D s (cid:105) . (2 b )Here U AD plays the same role and can have the samestructure as U SA .Note that Eqs. (2 a,b ) implement repeatable measure-ment / copying on the states {| s (cid:105)} , {| A s (cid:105)} of the systemand of the apparatus, respectively. That is, these statesof S and A remain untouched by the measurement andcopying processes. Repeatability implies that the out-come states {| s (cid:105)} as well as the record states {| A s (cid:105)} areorthogonal [13, 14]. This will matter in our discussion ofmeasurements involving mixtures.When the information about the outcome is copied,the combined pre-measurement state (cid:0)(cid:80) s α s | s (cid:105) (cid:1) | A (cid:105) of SA pair cannot be restored by U †SA . That is: U †SA (cid:0)(cid:88) s α s | s (cid:105)| A s (cid:105)| D s (cid:105) (cid:1) = | A (cid:105) (cid:0)(cid:88) s α s | s (cid:105)| D s (cid:105) (cid:1) (3 b )The apparatus is restored to the pre-measurement | A (cid:105) ,but the system remains entangled with the memory de-vice. On its own, its state is represented by the mixture: (cid:37) S = (cid:88) s w ss | s (cid:105)(cid:104) s | (7)where w ss = | α s | . Reversing quantum measurement ofa state that corresponds to a superposition of the poten-tial outcomes is possible only providing the memory ofthe outcome is no longer preserved anywhere else in theUniverse. C. Quasiclassical case
The special (measure zero) case when the quantumsystem is, prior to the measurement, in the eigenstate ofthe measured observable, constitutes an interesting ex-ception to the above “impossibility to reverse”. Thenthe measurement outcome: | s (cid:105)| A (cid:105)| D (cid:105) U SA = ⇒ | s (cid:105)| A s (cid:105)| D (cid:105) (1 c )can be copied | s (cid:105)| A s (cid:105)| D (cid:105) U AD = ⇒| s (cid:105)| A s (cid:105)| D s (cid:105) (2 c )and yet the evolution of SA can be reversed. | s (cid:105)| A s (cid:105)| D s (cid:105) U †SA = ⇒ | s (cid:105)| A (cid:105)| D s (cid:105) (3 c )The above three equations describe evolution of quantumsystems, yet they have the same structure and allow forthe reversal in spite of the record retained by D in thesame way as for the classical case (motivating the use of“quasiclassical” in the title of this subsection).It is straightforward to show that the same conclu-sion holds for mixed states that are diagonal in the ba-sis in which the system is measured. That is, the pre-measurement ρ S = (cid:80) s w ss | s (cid:105)(cid:104) s | is then identical as thepost-measurement (cid:37) S where the ”pre” and ”post” areindicated in using different version of Greek “rho”.This mixed quasiclassical case parallels classical Eqs.(4 a -6 a ). D. Superpositions of Outcomes and MeasurementReversal
We have now demonstrated the difference between thein principle ability to reverse quantum and classical mea-surements. Information flows do not matter for classical,Newtonian dynamics. However, when information abouta quantum measurement outcome is communicated—copied and retained by any other system—the evolutionthat led to that measurement cannot be reversed. Thus,from the point of view of the measurer, information re-tention about an outcome of a quantum measurementimplies irreversibility.We have also examined the quasiclassical case and con-cluded that the presence of arbitrary superpositions inquantum theory is responsible for the irreversibility ofmeasurements: When the considerations are restricted tosuch a quasiclassical set of orthogonal states, reversibilityof measurements is restored. Physical significance of thephases between the potential outcomes makes quantumstates vulnerable to the information leakage and preventsreversal of the evolution that led to the measurement.This significance of arbitrary superposition was illus-trated by the example of a mixture diagonal in the set ofstates that is left unperturbed by measurements. Mea-surement on a mixture that is diagonal in the same basiswith which measurements correlate the state of the appa-ratus remains in principle reversible. Thus, in a quantumUniverse where measurements are carried out only onpre-decohered systems (e.g., macroscopic systems in ourUniverse) and observers acquire information only aboutthe decoherence-resistant states, one may come to be-lieve that reversible dynamics is all there is. Of course,decoherence is an irreversible procsess, so in a sense, inour Universe, the price for this illusion of Newtonian re-versibility is a massive irreversibility which is paid “upfront”, extracted by decoherence.Presence of superpositions in correlated states of quan-tum systems can be quantified by quantum discord [15–17]. We shall now examine the relation between quantumdiscord and the ability to reverse measurements.
III. MEASUREMENTS OF QUANTUMMIXTURES, REVERSIBILITY, AND DISCORD
The above conclusion about the impossibility to re-verse quantum measurements (except for the quasiclassi-cal case) continues to apply when the pre-measurementstate of the system is a mixture diagonal in a basis thatis different from the measurement basis {| s (cid:105)} definedby U SA = (cid:80) s,k | s (cid:105)(cid:104) s || A k + s (cid:105)(cid:104) A k | . Thus, when the pre-measurement density matrix of the system is given by: ρ S = (cid:88) r,s w rs | r (cid:105)(cid:104) s | , (8)measurement by A results in a combined state: (cid:0)(cid:88) r,s w rs | r (cid:105)(cid:104) s | (cid:1) | A (cid:105)(cid:104) A | U SA = ⇒ (cid:88) r,s w rs | rA r (cid:105)(cid:104) sA s | . (9)Copying: (cid:0)(cid:88) r,s w rs | rA r (cid:105)(cid:104) sA s | (cid:1) | D (cid:105)(cid:104) D | U AD = ⇒ (cid:88) r,s w rs | rA r D r (cid:105)(cid:104) sA s D s | (10)leads to a state that exhibits quantum correlations be-tween all three systems. Reversal of the evolution: (cid:88) r,s w rs | rA r D r (cid:105)(cid:104) sA s D s | U †SA = ⇒| A (cid:105)(cid:104) A | (cid:88) r,s w rs | rD r (cid:105)(cid:104) sD s | (11)that acts purely on the SA pair restores only the pre-measurement state of the apparatus, but not the state ofthe system, (cid:37) S = (cid:88) s w ss | s (cid:105)(cid:104) s | = Tr D ( (cid:88) r,s w rs | rD r (cid:105)(cid:104) sD s | ) , (12)as the reduced density matrix of the system is now—unlike the pre-measurement ρ S , Eq. (8)—diagonal in themeasurement basis | s (cid:105) .Thus, in contrast to the classical case, acquiringand communicating information about quantum systemsmatters: Reversibility of the global dynamics is notenough. Presence of a copy of the information (that didnot matter in the classical case) precludes the possibilityof implementing local reversals.The information-theoretic price—the extent ofirreversibility—can be quantified by ∆ H , the differ-ence in entropy between the pre-measurement andpost-measurement density matrices;∆ H = − (cid:88) s w ss lg w ss + Tr ρ S lg ρ S = H ( (cid:37) S ) − H ( ρ S ) . (13)We shall now show that this entropy increasecaused by copying coincides with the quantum dis-cord [15–17] in the correlated post-measurement state (cid:80) r,s w rs | rA r (cid:105)(cid:104) sA s | of the system and the apparatus.This suggests that vanishing of discord may be a con-dition for the reversibility undisturbed by copying. A. Introducing quantum discord
Discord is the difference between the mutual informa-tion defined by the symmetric equation that involves vonNeumann entropies of the two systems separately andjointly: I ( S : A ) = H S + H A − H SA , (14)where H X = − Tr ρ X lg ρ X , and the asymmetric definitionof mutual information J ( S ; A ) A|{| A k (cid:105)} . The asymmetric version of mutual information obtainsfrom the joint entropy when it is expressed in terms ofthe conditional entropy: H SA|{| A k (cid:105)} = H S|A{| A k (cid:105)} + H A|{| A k (cid:105)} , (15)where we have assumed that the measurements were per-formed on A in the basis {| A k (cid:105)} . Thus, H A|{| A k (cid:105)} is theentropy computed using probabilities of states {| A k (cid:105)} ,and H S|A{| A k (cid:105)} is the conditional entropy one still hasafter the outcomes of measurement on A in the basis {| A k (cid:105)} are known.In the classical setting, when Shannon entropies arecomputed from classical probabilities, analogous two ex-pressions for the joint entropy coincide [18]. However, inthe quantum setting, possible post-measurement states—hence, conditional information—have to be defined withrespect to the basis set characterizing the measurementthat is carried out on one of the two systems (here A ) inorder to gain partial information about the other (here S ). Using this basis-dependent joint entropy H SA|{| A k (cid:105)} in Eq. (14) instead of H SA one gets an asymmetric ex-pression for mutual information: J ( S ; A ) A|{| A k (cid:105)} = H S + H A − ( H S|A{| A k (cid:105)} + H A|{| A k (cid:105)} ) . (16)Discord is the difference between the symmetric andasymmetric formulae for mutual information : δ ( S : A ) A|{| A k (cid:105)} = I ( S : A ) − J ( S ; A ) A|{| A k (cid:105)} , (17 a )or; δ ( S : A ) A|{| A k (cid:105)} = ( H S|A{| A k (cid:105)} + H A|{| A k (cid:105)} ) − H SA . (17 b )When the two systems are classical (so that their statescan be completely described by probabilities) the two def-initions of the mutual information coincide, and quantumdiscord disappears—it is identically equal to zero. In thequantum domain probabilities usually do not suffice, andthe two expressions for the mutual information differ.In the case we have considered above the system wasin a mixed state, but the initial state of the appara-tus was pure, and the measurement that correlated S with A was unitary, so that H SA = H ( ρ S ). Moreover, H S|A{| A s (cid:105)} = 0 (as a measurement of A with the re-sult | A k (cid:105) reveals the corresponding pure states of S ) and H A|{| A s (cid:105)} = H ( (cid:37) S ) (as the entropy of A is, after it cor-relates with S computed from the probabilities w ss andequals − (cid:80) s w ss lg w ss ). Consequently, the entropy in-crease ∆ H of Eq. (13) is indeed equal to the discord inthe post-measurement (but pre-copying) state of SA . There are subtleties in the definition of the discord. Definitiongiven here is the so-called thermal discord or one-way deficit. Itdiffers from the “original” discord defined in [15–17]. A briefdiscussion in the context of Maxwell’s demon can be found in[19]. More extensive discussions of discord and related measuresare also available [20, 21]. We note that appearance of discordin the correlated SA state can be traced [22] to the presence ofquantum coherence in the states of S . B. Reversibility and quantum discord
We now consider a general case, where the pre-measurement density matrices ρ S , ρ A and the post-measurement ρ SA can all be mixed. The evolution thatleads to the measurement is still unitary U SA . And westill assume that the apparatus should obtain and retainat least an imperfect record of the system. That is, thereshould be states { ρ S s } of the systems that leave imprintson the state of the apparatus: ρ S s ρ A U SA = ⇒ ρ SA s . (18)An initial mixture of { ρ S s } will evolve, by linearity, intothe corresponding mixture of the outcomes. (cid:88) s p s ρ S s ρ A U SA = ⇒ (cid:88) s p s ρ SA s = ρ SA . (19)The correlation could be imperfect (i.e., one might onlybe able only infer some information about some of the { ρ S s } from A ).Copying involves interaction of A and D . As before,we enquire under what circumstances transfer of infor-mation about S via A to D does not preclude reversal,so that the evolution generated by U †SA restores the pre-measurement state of SA in spite of the correlation with D established by: (cid:88) s p s ρ SA s | D (cid:105)(cid:104) D | U AD = ⇒ (cid:88) s p s ρ SA s | D s (cid:105)(cid:104) D s | = ρ SAD (20)To allow for reversal the state of SA must not be affectedby the copying. That is, (cid:37) SA = Tr D ρ SAD = ρ SA , (21)where ρ SA and (cid:37) SA are the density matrices beforeand after the copying operation. This is a density ma-trix version version of the “repeatability condition” (see[13, 14]): Copying can be repeated (since the “original”remains unchanged), and we shall see that this repeata-bility leads to similar consequences—to the orthogonalityof the records that can be copied.Unitarity of U AD is responsible for our next re-sult. Unitary evolutions preserve Hilbert-Schmidt norm.Therefore, (cid:88) r,s p r p s Tr ρ SA r ρ SA s = (cid:88) r,s p r p s Tr ρ SA r ρ SA s |(cid:104) D r | D s (cid:105)| . (22)The overlap of the copy states in D is non-negative andbounded, 0 < |(cid:104) D r | D s (cid:105)| ≤
1. Therefore, there are onlytwo ways to satisfy this equality: Either |(cid:104) D r | D s (cid:105)| = 1(i.e., there is no copy!), or p r p s Tr ρ SA r ρ SA s = 0 . (23)For the non-trivial case when p r p s > r (cid:54) = s thisleads to; Tr ρ SA r ρ SA s = 0 (24) as a necessary condition to allow for copying that doesnot interfere with the possibility of the reversal.Indeed, when (as we have assumed) copying evolutionoperator U AD involves only A and D , we can repeat theabove reasoning starting with the reduced density matrixof A alone and demanding that it is untouched by thecopying operation: (cid:37) A = Tr SD ρ SAD = ρ A . (25)(Clearly, if copying were to affect density matrix of A , itwould affect also ρ SA , so Eq. (21) cannot not be satisfiedunless Eq. (25) holds.)In the end we will conclude that repeatability is notruled out by retention of the copies of the outcomes pro-viding that: p r p s Tr ρ A r ρ A s = 0 . (26)For the non-trivial case when p r p s > ρ A r ρ A s = 0 (27)as a necessary condition to allow for copying of the infor-mation from A that does not interfere with the possibilityof the reversal.To assure that copying will indeed leave (cid:37) SA un-changed, we need satisfy the same condition that se-lects pointer states [23, 24]: The unitary U AD that pro-duces copies must commute with the pre-copying (cid:37) SA to leave it unaffected. This will be the case when theHamiltonian H AD that generates U AD commutes withthe pointer observable of A —with the apparatus observ-able that keeps the records of the state of the system.This pointer observable will have in general degenerateeigenstates—eigenspaces that serve (within the appara-tus Hilbert space) as a “one leg” of the support of thedensity matrices ρ SA s . Orthogonality of the record statesof A implies zero “one way” discord in the basis corre-sponding to these pointer eigenspaces.We note that there is an important difference betweenEqs. (23, 24) and Eqs. (26, 27) we have derived. Theyrely on different assumptions: Eqs (26, 27) are “local” –they focus on the content of the records in the apparatusalone, and demand distinguishabiility (orthogonality) ofits states. This focus is justified by the nature of thecopying interaction—it involves only A and D , so onlythe records in A are relevant. By contrast, Eqs. (23, 24)could be satisfied equally well by orthogonality of localstate of S alone or, indeed, of the global states of SA . Inother words, when one can access the composite system AS , the condition that allows for reversible copying canbe satisfied by the global state even when it is not metby the record states of A alone [14]. Our next goal is toconsider effects of such more global copying operations. IV. KNOWING OF THE RECORD BUT NOTTHE OUTCOME
Immediately above, in Eqs. (25-27), we have insistedthat the orthogonality condition Tr ρ SA r ρ SA s = 0 shouldbe satisfied “in the apparatus”, that is, that the appara-tus eigenspaces that correspond to the records should beorthogonal. This insistence stemmed from the fact thatthe copying evolution U AD coupled only to A . However,one can imagine a situation where U ( SA ) D couples D to aglobal observable of SA . In that case, one might be ableto find out that A “knows” the outcome – the state of S – without actually finding out the outcome.The simplest such example is afforded by a one qubitapparatus that measures a one qubit system. Thecorrelated—entangled—state of the two is then simply: | ψ SA (cid:105) = a ↑ | ↑ A ↑ (cid:105) + a ↓ | ↓ A ↓ (cid:105) (28)in obvious notation. Agent D can then detect presenceof the correlations established when S and A interacted.We now consider two operators that can confirm theexistence of the correlation between S and A . The firstsuch operator, when measured, would establish whetherthe states of S and A are correlated in the basis (here {| ↑(cid:105) , | ↓(cid:105)} ) in which the measurement was carried out: ˆA = y ↑ | ↑ A ↑ (cid:105)(cid:104)↑ A ↑ | + y ↓ | ↓ A ↓ (cid:105)(cid:104)↓ A ↓ | + n | ↑ A ↓ (cid:105)(cid:104)↑ A ↓ | + n (cid:48) | ↓ A ↑ (cid:105)(cid:104)↓ A ↑ | . (29)The detection of either of the y eigenvalues would im-ply a successful measurement (while either of n eigenval-ues would signify error). Moreover, when y ↑ = y ↓ = y ,such measurement would reveal consensus without be-traying the actual outcome. Thus, agent D —friend ofthe observer—could confirm the success of the measure-ment, but the evolution that led to the measurement canbe be still undone.This is “relative reversibility”—the evolution that ledto measurement can be at least in principle undone by anagent who can confirm that the measurement was suc-cessful providing he does this without finding out theoutcome. When y ↑ (cid:54) = y ↓ , the measurement by D wouldcorrelate his state with the outcome, and the reversalwould become impossible.An alternative confirmation of a successful SA mea-surement can be accomplished by detecting entanglementin | ψ SA (cid:105) . Bell operator: ˆB = b += | β += (cid:105)(cid:104) β += | + b − = | β − = (cid:105)(cid:104) β − = | + b + (cid:54) = | β + (cid:54) = (cid:105)(cid:104) β + (cid:54) = | + b −(cid:54) = | β −(cid:54) = (cid:105)(cid:104) β −(cid:54) = | (30)can be used for this purpose. Above, subscripts “=” and“ (cid:54) =” stand for “parallel” and “antiparallel”, and the Belleigenstates are; | β ± = (cid:105) = | ↑ A ↑ (cid:105) ± | ↓ A ↓ (cid:105) ; (31 a ) | β ±(cid:54) = (cid:105) = | ↑ A ↓ (cid:105) ± | ↓ A ↑ (cid:105) . (31 b ) Detection of either b += or b − = implies successful measure-ment. However, unless b += = b − = , measurement will alsoreveal phases between the outcome states, and (unless | ψ SA (cid:105) happens to be one of the above Bell states) it willresult in decoherence in the Bell basis (and, hence, pre-vent reversal).It is interesting to note that when one imposes degen-eracy that enables reversal on either ˆA or ˆB , eigenstatesof these two operators coincide. The resulting consensusoperator is given by: ˆC = y ( | ↑ A ↑ (cid:105)(cid:104)↑ A ↑ | + | ↓ A ↓ (cid:105)(cid:104)↓ A ↓ | )+ n ( | ↑ A ↓ (cid:105)(cid:104)↑ A ↓ | + | ↓ A ↑ (cid:105)(cid:104)↓ A ↑ | ) (32)Thus, by measuring ˆC one can confirm that A “knows theoutcome” without impairing the possibility of reversal. V. DISCUSSION
Our results shed new light both on the relation be-tween quantum and classical and on the role of informa-tion in measurements. So far we have mainly emphasizedtheir relevance for the distinction between quantum andclassical physics. To re-state briefly the main conclu-sion, retention of information about classical states hasno bearing on the in principle ability to reverse classicalevolution that leads to measurement, but it precludes re-versing quantum measurements (with the exception ofthe quasiclassical case). Thus, information plays a farmore important role in quantum Universe than it usedto play in classical physics.This operational view of reversibility yields new in-sights:(i) In quantum physics irreversibility in course of mea-surements need not be blamed solely on decoherence, butis caused by observer’s acquisition of the data about thesystem. Observer who retains record of the outcome can-not restore the pre-measurement states of both the sys-tem S and the apparatus A . So, from observer’s point ofview, while classical measurements can be undone, quan-tum measurements are fundamentally irreversible.(ii) Acquisition of information results in decrease ofthe von Neumann entropy of the system. Therefore, thisaspect of irreversibility of measurements is not a conse-quence of the second law. Yet, while observer can takeadvantage of this (apparent!) violation of the second law,he cannot reverse measurement on his own.(iii) However, observer’s friend (who knows about themeasurement, but not its outcome) can, in principle, in-duce such a reversal providing there is no copy of therecord of the outcome left anywhere.Our discussion calls for a re-consideration of the natureand origin of the initial conditions in quantum physics.Distinction between the laws that dictate evolution ofthe state of a system and initial conditions the define itsstarting point dates back to Newton [8]. This clean sep-aration is challenged by quantum measurements. Seenfrom the inside, by the observer, measurement re-setsinitial conditions. Acquisition of information simultane-ously redefines the state of the observer and observer’sbranch of the universal state vector. From then on, ob-server will exist within the Universe he helped define (seeFig. 1). On the other hand, observer’s friend will—foras long as he does not find out what the observer foundout—live in a Universe where the initial condition is thepre-measurement state with a coherent superposition ofall the potential outcomes.Familiar “paradox” of Wigner’s friend offers an inter-esting setting for this discussion. Wigner speculated [9](following to some extent von Neumann [1]) that “col-lapse of the wavepacket” may be ultimately precipitatedby consciousness. The obvious question is, of course,“how conscious should the observer be”. The answer suggested by our discussion is that—if theevidence of collapse is the irreversibility of the evolu-tion that caused it—retention of the information suffices.Thus, there is no need for “consciousness” (whatever thatmeans): Record of the outcome is enough. On the otherhand, observer conscious of the outcome certainly retainsits record, so being conscious of the result suffices to pre-clude the reversal—to make the “collapse” irreversible.Quantum Darwinism [11] traces emergence of the ob-jective classical reality to the proliferation of informa-tion throughout the environment. Our discussion of theconsequences of retention of information for reversibilityis clearly relevant in this context, although its detailedstudy is beyond the scope of this paper.This research was supported by the Department of En-ergy via LDRD program in Los Alamos, and, in part, bythe Foundational Questions Institute grant “Physics ofWhat Happens”. [1] von Neumann, J. Mathematical Foundations of QuantumTheory , translated from German original by R. T. Beyer(Princeton University Press, Princeton, 1955).[2] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch andI.-O. Stamatescu
Decoherence and the Appearance of aClassical World in Quantum Theory (Springer-Verlag,Heidelberg, 2003).[3] W. H. Zurek,
Rev. Mod. Phys. , 715 (2003).[4] M. Schlosshauer, Decoherence and the Quantum to Clas-sical Transition , (Springer, Berlin, 2007).[5] W. H. Zurek,
Physics Today , 44 (2014).[6] Zeh, H D., Physical Basis of the Direction of Time ,(Springer, Berlin, 2001).[7] Zurek, W. H., and Paz, J.-P.,
Phys. Rev. Lett. , 2508(1994).[8] E. P. Wigner, Nobel Lecture, (1963); “ Thesurprising discovery of Newton’s age is justthe clear separation of laws of nature on theone hand and initial conditions on the other. ” − prizes/physics/laureates/ /wigner − lecture.pdf [9] E.P. Wigner (1961), “Remarks on the mind-body ques-tion”, in I.J. Good, The Scientist Speculates , (London,Heinemann, 1961).[10] J. A. Wheeler, (1983), “Law without Law”, in J. A.Wheeler and W. H. Zurek,
Quantum Theory and Mea-surements , (Princeton University Press, Princeton,1983).[11] H. Ollivier, D. Poulin, and W. H. Zurek,
Phys. Rev.Lett. , 220401 (2004); R. Blume-Kohout and W. H.Zurek, Phys. Rev. A , 062310 (2006); J. P. Paz andA. J. Roncaglia, Phys. Rev. A , 042111 (2009); W. H.Zurek, Nature Physics , , 181 (2009); C. J. Riedel andW. H. Zurek, Phys. Rev. Lett. , 020404 (2010):
N. J.Phys. Sci.Rep. , 1729 (2013); M. Zwolak, C. J. Riedel, and W. H.Zurek, Phys. Rev. Lett. , 140406 (2014); J. K. Kor-bicz, P. Horodecki, and R. Horodecki,
Phys. Rev. Lett. , 120402 (2014); F. G. S. L. Brandao, M. Piani, andP. Horodecki,
Nat. Comm. , 7908 (2015). C. J. Riedel,W. H. Zurek, and M. Zwolak, Phys. Rev. A , 032126 (2016).[12] N. Bohr, Nature , 580 (1928).[13] W. H. Zurek,
Phys. Rev. A , 052110 (2007).[14] W. H. Zurek, Phys. Rev. A , 052111 (2013).[15] W. H. Zurek, Ann. Phys. Leipzig , 855 (2000).[16] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. , , 017901(2001).[17] L. Henderson and V. Vedral, J. Phys. A, , 6899 (2001).[18] T. M. Cover and J. A. Thomas, Elements of InformationTheory (Wiley-Interscience, New York, 2006).[19] W. H. Zurek
Phys. Rev. A, , 012320 (2003).[20] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V.Vedral, Rev. Mod. Phys. , 1655 (2012).[21] A. Bera, T. Das, D. Sadhukhan, S. S. Roy, A. Sen(De),U. Sen, Rep. Progr. Phys. (2018).[22] J. Ma, B. Yadin, D. Girolami, V. Vedral, and M. Gu Phys. Rev. Lett. , 160407 (2016).[23] W. H. Zurek,
Phys. Rev. D , 1516 (1981).[24] W. H. Zurek, Phys. Rev. D26