24 October 1994

PHYSICS LETTERS A

ELSFVIER Physics Letters A 194 ( 1994) 2 l-25

Time asymmetry in quantum mechanics: a retrodiction paradox Asher Peres 1,2

Theoretical Physics Institute, University of Alberta, Edmonton, Canada T6G 251

Received 16 May 1994; accepted for publication 7 September 1994 Communicated by P Holland

Abstract

The knowledge of the quantum state of a physical system enables us to make verifiable statistical predictions on the future behavior of that system, as long as its Hamiltonian is known. On the other hand, retrodictions of past properties are not verifiable, and may lead to inconsistencies for correlated systems.

In the early days of quantum mechanics, Einstein, Tolman, and Podolsky [ 1 ] argued that “quantum me- chanics must involve an uncertainty in the description of past events which is analogous to the uncertainty in the description of future events.” That is, quantum me- chanics not only precludes the exact prediction of the results of future measurements, but also the exact re- construction of the past history of a physical system. It was later shown [ 21 that there appeared to be a formal symmetry between past and future in the measuring process. These results have been the basis of a “two- time” quantum formalism [ 31 which treats prediction and retrodiction in a symmetric way. The purpose of this note is to show that, contrary to these claims, or- thodox quantum mechanics is not time symmetric and that naive retrodiction may lead to inconsistencies.

Time-symmetry has also been postulated in some variants of quantum mechanics, such as the many world interpretation [ 41 or Griffiths’s “consistent history” approach [ 51. The latter uses the standard formulation of quantum mechanics, supplemented

’ Permanent address: Department of Physics, Technion - Israel Institute of Technology, 32 000 Haifa, Israel. 2 E-mail: peres@techunix.technion.ac.il.

Elsevier Science B.V. SSDI 0375-9601(94) 00744-6

by non-standard interpretation postulates: a modified Born probability rule “puts predictions and retrod- ictions on an equal footing, but does it at the price of ruling out certain inferences allowed by classical logic and probability theory” [6]. In this Letter, I shall adhere to the conventional “orthodox” quantum formalism: it is the only one which is actually used by experimental physicists for analyzing the results recorded by their macroscopic instruments [ 71.

The origin of time asymmetry in conventional quan- tum mechanics is that the measuring process itself is fundamentally time-asymmetric: ideally, it produces a correlation between the final states of the quantum system and those of the measuring apparatus, while these states were uncorrelated before the measure- ment. Therefore an ideal (i.e., repeatable, von Neu- mann type) measurement of a complete set of com- muting operators also is the preparation of a pure quantum state. After that preparation, we can predict with certainty the results of some future measure- ments, namely those of operators having that quantum state as one of their eigenstates. These “nondemoli- tion” measurements can be performed without altering the state of the quantum system. Since their result is

22 A. Peres /Physics Letters A 194 (1994) 21-25

known with certainty, we may consider this result as objectively true (even if the measurement is not actu- ally performed) provided that the required preparu- tion of a known pure state has actually been carried out. The last condition is crucial: it was not imposed by Einstein, Podolsky, and Rosen [ 81 when they de- fined their “elements of reality,” and this is what led them to a paradox.

On the other hand, in the absence of extraneous information restricting the choice of possible initial states, the result of a quantum measurement does not completely specify the state of the quantum system before that measurement: it only supplies a posteriori probabilities for that state. In particular, the quantum state before the measurement cannot be orthogonal to the one that was actually observed (this property will be used later).

The state of a quantum system can also be deter- mined indirectly, without interacting in any way with that system, by measuring another, distant system, which has been initially correlated to the first one [ 81. The knowledge of that state can then be used for pre- dictions, as usual; but, as shown below, retrodictions may be inconsistent.

The quantum system that I shall discuss here is the familiar pair of spin 4 particles, prepared in a singlet state and moving apart from each other, toward two distant observers. To shorten the discourse, these will be called Alice and Bob, as usual in quantum commu- nication problems [ 9, IO] . A large number of quantum pairs are thus prepared by a third party, half way be- tween the observers. These pairs are fully distinguish- able - for example, they are produced at the rate of one per day - so that the particles received by each observer can be unambiguously matched pairwise. It is further assumed that these particles do not inter- act with each other or with any external agent from the moment they have been produced in the singlet state (their Hamiltonian is H = 0). The role of the observers is to ascertain the nature of the preparation state.

When Alice receives a particle, she decides in a random way whether to measure the x-component of its spin, or to store that particle for later use. She then informs Bob of what she has done and found. Likewise, Bob randomly decides whether to measure the y-component of the spin of any particle that he receives, or to store it for further use, and he informs

Alice of what he is doing. The particle pairs treated in this way thus fall into four sets, which will be labelled YY, YN, NY, and NN, depending on whether Alice and/or Bob performed a spin measurement.

Note that the situationdescribed above is fundamen- tally different from the one leading to the Einstein- Podolsky-Rosen paradox [ 81: the choice given to each observer is not between incompatible measurements (some of which are necessarily counterfactual), but between performing or not performing a well defined measurement. All procedures are known and open to verification. Both observers are aware of each other, and no one is cheating or concealing any information. as in cryptographic setups [ 9, IO]. The only peculiar- ity here is that each observer receives the particles be- fore receiving the corresponding information from the other observer.

Consider first the particles of the NN set. If none of the particles of a given pair is measured by Alice and Bob, these two particles can be brought together by a process which does not modify their spin state. in order to test whether the pair indeed is in a singlet state. For instance, the n-component of total spin, J, , may be measured in a repeatable way which does not affect J, nor J*. The result should always be J,V = 0. And thereafter JY is measured and should likewise be found null. If this double test is performed on a single pair, it does not prove as yet that the initial state was a singlet; it merely proves that it was not a pure triplet, because, for total J = 1, the two states 1 J, = 0) and / JY = 0) are mutually orthogonal. Only this restricted retrodiction is legitimate, as explained earlier in this Letter. However, if numerous identically prepared pairs consistently give the same null result, it may be inferred with probability close to 1 that they were in a pure singlet state.

Next, we turn our attention to the particles of the YN set. For each one of them, Bob has received from Alice the information whether she found (TAX = I or -- 1. He can test VnX for some of these particles, randomly chosen, to check that the information sent by Alice is reliable: he should always find the opposite result because, for a singlet state p, quantum mechanics predicts that the mean value of ~~~~~~ is

(~/~A#JB.xI~) =-I. (11

This relation, which is a consequence of Born’s prob- ability rule, guarantees that measurements of (TV on

A. Peres /Physics Letters A 194 (1994) 21-25 23

both particles always yield opposite results, fl and T 1. Note that this property holds even if the measure- ments are not of the von Neumann type (that is, re- peatable, and with the wave function collapsing into an eigenstate of the operator that was measured). Any other type of projection valued measure [ 111 satisfies Rq. (1) and is admissible.

Having convinced himself of the reliability of Al- ice’s information by testing a large number of ran- domly chosen samples, Bob can now predict with cer- tainty the reSdtS of measurements of (Tax on the re- maining particles of the YN set: each one of them is in a known eigenstate of (7nX . Moreover, Bob may also be tempted to retrodict that each one of these parti- cles was in the same eigenstate of (+nx even before he received Alice’s information (because Bob is a real- ist and he cannot admit that it is the arrival of Alice’s message which determines the physical state of a par- title) . Stated more precisely, if Bob assumes that the state of each one of these particles, namely Icl,’ or I& (satisfying crBX& = %I,@ ) , is an objective property of the particle and does not change with time (be- cause H = 0), the inescapable inference is that the particle had the same state all along its world line, from the moment it ceased to interact with its partner. Likewise, Alice may be tempted to retrodict that the particles of the NY set were in definite eigenstates of ffAy , even before she received Bob’s information, be- cause, if she has not yet tested a particle when that information arrives, she will always be able to verify that Bob’s information was correct.

However, these two retrodictions, for particles of the YN and NY sets, are incompatible with the veri- fiable properties of a singlet state that were discussed above. For example, let us assume that each particle of the YN set is received by Bob with a definite value of an, (either + 1 or - 1) which is merely unknown to him, until Alice’s message arrives. In that case, each particle received by Alice must also have a definite Value of (TAX, opposite to that of CrnX (it has to be so, as otherwise Bob would not consistently find the result opposite to the one stated in Alice’s message, whenever he decides to check the latter). Therefore, if retrodiction is a legitimate way of reasoning, the par- ticles of the YN set behave as if they were produced in a factorable state $zqQg , where I,@ and fiz are eigenstates of (TAX and an,, respectively, with oppo- site eigenvalues.

Such a state indeed satisfies J, = 0, which is the only property actually tested for the YN particles. On the other hand, $z@ is not a singlet: that state has observable statistical properties different from those of a pair of particles of the NN set [ 121. It is so even before Alice performs her measurement of (7AX, that is, before any measurement has been done. For example, the observed value of

Jy2 = +( ff.4~ + gBy )* = ; + $~~A~DB~ , (2)

cannot always vanish, as it would do for a singlet. Indeed, for a factorable state, we have on the average,

((+A? (+By) = ((+A?) @By) = 0, (3)

because any @ which is an eigenstate of a, satis- fies ($lay]$) = 0. It then follows from Eq. (2) that ( Jy2) = i. We thus see that observable statistical prop- erties of the NN set (namely, (Jy2) = 0), are differ- ent from those of the identically prepared YN set, be- fore any measurement is performed on the latter (ifit is assumed that the YN set obeys Bob’s retrodiction, namely that the state +n of each particle received by him is an eigenstate of I+nx) .

Where is the contradiction? Recall that all the par- ticle pairs were prepared according to the same pro- cedure, and then divided into four subsets in a ran- dom way, before some of them were measured. It is therefore reasonable to assume that these four sub- sets initially had identical statistical properties (the alternative would be to assume that the mere inten- tion of measuring some particles would affect the state in which they are produced). However, the verifiable result predicted above for all the pairs of type NN, namely Jy = 0, is incompatible with properties of the YN set that can be derived from a retrodiction argu- ment. Therefore, if retrodiction is taken seriously, it inevitably leads to the extravagant conclusion that the initial properties of the YN and NN sets are different, in spite of their identical preparations, merely because Alice intends to measure the particles of the YN set!

The asymmetry between prediction and retrodiction is related to the fact that predictions can be verified (or falsified) by actual experiments, retrodictions can- not. Retrodictions are counterfactual statements about unper$onned experiments; in quantum mechanics, un- performed experiments have no results [ 131. In gen- eral, these retrodictions are harmless - because they

24 A. Peres /Physics Letters A 194 (1994) 21-25

Fig. 1. In this spacetime diagram, the origins of the coordinate systems are the locations of the two tests. The 1~ and ta axes are the world lines of the observers, who are receding from each other. In each Lorentz frame, the zA and zg axes are isochronous: fA = 0 and tg = 0, I%pectively.

are not experimentally verifiable or falsifiable - but, in the present case of entangled states, they lead to an inconsistency.

Finally, consider the YY pairs, for which both (TAX and @a, are measured. In a nonrelativistic theory, where information transfer can be instantaneous, it may be argued that each measurement “collapses” the wave function of both particles. I shall not discuss here the notorious difficulties related to the collapse pos- tulate, but only recall that the latter is incompatible with Lorentz-invariance [ 141. To illustrate this point, let our two observers be attached to different Lorentz frames, as shown in Fig. 1. They recede from each other, with a constant relative velocity. Then, in each one of the Lorentz frames, the test performed by the observer who is at rest appears to occur earlier than the test performed by the moving observer.

Thus, if Alice believes that the pair of particles has, at each instant tA , a definite wave function, she will say that the singlet state, which existed for tA < 0, collapsed into an eigenstate of flux for fA > 0. In the same vein, Bob may say that the singlet state held as long as ta < 0, and thereafter collapsed into an eigen- state of PAy as a result of his test. Such statements are not only contradictory - they are utterly meaning- less. There is no disagreement about what was actually observed, but a situation involving several observers cannot be described by a single wave function with a relativistic transformation law.

What would be the price of imposing time- symmetry on the conventional “orthodox” formula- tion of quantum mechanics (the one with the familiar Born probability rule) ? This issue was analyzed by Costa de Beauregard [ 151 who came to the conclu- sion that “intrinsic time symmetry implies intrinsic symmetry between retarded and advanced waves,” and that “relations between distant systems are prop- agated both forward and backward in time.” For example, the information flowing back from Alice’s measurement would destroy the initial singlet state and replace tt by $A (GB ?? ‘f , which is an eigenstate of (TAX and flnx. The implications are clear: propagating information backward in time has the same meaning as rewriting history. This indeed is current practice in some political contexts, but most physicists are reluctant to seriously consider such radical solutions.

On the other hand, no contradiction arises if we refrain from attributing to the state vector 1+5 an on- tic meaning, and we consider ti as a mere mathemat- ical tool, representing information that can be used for computing probabilities of results of future mea- surements [ 71. With that interpretation of @, Bob’s particle acquires a definite state if and when Bob re- ceives Alice’s message (because $ has no other mean- ing than an input needed by Bob for predicting future events). This approach may appear unsatisfactory to some readers. The purpose of this Letter was to ex- pose the consequences of its possible alternatives.

A. Peres /Physics Letters A 194 (1994) 21-25 25

I am grateful to Richard Jozsa for raising questions which led to this work, and to Don Page for many interesting comments.

References

[ 11 A. Einstein, R. C. Tolman and B. Podolsky, Phys. Rev. 37 (1931) 780.

[2] Y. Ahaconov, R G. Bergmann and J. L. Lebowitz, Phys. Rev. 134 (1964) Bl410.

[3] Y. Aharonov and L. Vaidman, J. Phys. A 24 (1991) 231.5, and references therein.

[4] D. N. Page, Phys. Rev. Wt. 70 (1993) 4034.

[5] R. B. Griffiths, J. Stat. Phys. 36 (1984) 219. [6] R. B. Griffiths, Ann. NY Acad. Sci. 480 (1986) 512. [7] N. van Kampen, Physica A 153 (1988) 97. [ 81 A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 ( 1935)

777. [9] A. K. Eke& Phys. Rev. Lett. 67 (1991) 661.

[lo] C. H. Bennett, Phys. Rev. Lett. 68 (1992) 3121. [ 111 F’. Busch, P J. Lahti and P Mittelstaedt, The quantum theory

of measurement (Springer, Berlin, 1991) p. 9. [ 121 D. Bohm and Y. Aharonov, Phys. Rev. 108 (1957) 1070. [ 131 A. Peres, Am. J. Phys. 46 (1978) 745. [14] 1. Bloch, Phys. Rev. 1.56 (1967) 1377. [ 151 0. Costa de Beauregard, Nuovo Cimento B 42 ( 1977) 41;

51 (1979) 267.