fulltext

Conditional probabilities in quantum mechanics

from a time-symmetric formulation (*)

D. J. MILLER

School of Physics, The University of New South Wales - Sydney NSW 2052, Australia (ricevuto il 23 Settembre 1996; approvato il 25 Giugno 1997)

Summary. — An expression is proposed for a conditional (pseudo-)probability measure for quantum mechanics which preserves as far as possible the relationships among probabilities that are expected for a classical probability space. The latter is achieved by relaxing the restriction that conditional probabilities for counterfactuals must lie in the range [ 0 , 1 ]. The probabilities for verifiable properties are the same as in orthodox quantum mechanics but they are found without invoking the projection postulate. The new formalism is applied to the case of a spin-singlet pair and it is shown that the expressions for the conditional probabilities that appear in the Bell-Wigner inequalities sum to give the quantum-mechanical result. The proposed conditional pseudo-probability depends on the next relevant measurement as well as the preparation state and therefore it is a specific proposal for the transactional interpretation of quantum mechanics. The definition of the elements of reality must be revised to account for the possibility of probabilities outside the range [0, 1]. The elements of reality which can be assigned do not violate the product rule.

PACS 03.65.Bz – Foundations, theory of measurement, miscellaneous theories (including Aharonov-Bohm effect, Bell inequalities, Berry’s phase).

PACS 03.65.Ca – Formalism.

1. – Introduction

The non-commutative algebra of quantum mechanics means that many of the relationships among probabilities expected in classical probability spaces do not hold for quantum mechanics [1, 2]. For example, in the classical case, the conditional proba-bility p(qNs1S s2) that observable Q has the magnitude q conditional on two mutually exclusive counterfactual properties S1and S2having magnitudes s1and s2is given by

p(qNs1S s2) 4p(qNs1) p(s1) 1p(qNs2) p(s2) , (1)

where p(si) is the probability that Si has magnitude si, i 41, 2. This relationship does not hold for the conditional probabilities of quantum mechanics, for example in the two-slit experiment [3] where Siis the location at slit i and Q is the detection at a point distant from the slits.

(*) The author of this paper has agreed to not receive the proofs for correction.

The reason why the above relationship does not hold is that the conditional probability measure in orthodox quantum mechanics is assumed to be given by the Lüders rule [1, 2, 4, 5], the same rule that would have been used to calculate the projection of the density operator if the conditionalising event had involved an actual measurement. It is possible that a different conditional probability measure could apply to counterfactual events, like the passage through one or another of the slits in the above example, because in those cases no measurement takes place. In fact if a classical description of quantum systems between measurements is going to be able to be given, the conditional probabilities for counterfactuals involved in the description must be different from the Lüders rule. This is because any classical description would have to be based on the relationships expected in a classical probability space, but some of these do not follow from the Lüders rule. It has also been suggested that this problem can be overcome by excluding counterfactual properties from the treatment of probability in the quantum domain [6, 7].

The purpose of the present work is to introduce a conditional (pseudo-)probability measure for quantum mechanics which preserves the relationships expected for a classical probability space. In this approach the non-classical nature of the probability space manifests itself in that the probability for counterfactual properties of the quantum systems do not necessarily lie in the range [ 0 , 1 ]. For this reason the probabilities are referred to as pseudo-probabilities. One knows from the Bell-Wigner inequalities, for example, that positive values for the conditional probabilities for counterfactuals lead to disagreement with orthodox quantum mechanics. The probabilities calculated from the proposed formalism for verifiable properties, as opposed to counterfactual properties, are the same as in orthodox quantum mechanics.

Mückenheim [8] has reviewed the status of the use of negative probabilities in physical theories and Feynman [9] has concluded that conditional probabilities and probabilities of imagined intermediate states may be negative, or greater than one, in a calculation of the probabilities for physical events or states. The probabilities that lie outside the range [ 0 , 1 ] are used only in those ways in the present approach.

The conditional pseudo-probability measure that is proposed here can also be viewed as a specific proposal for the transactional interpretation of quantum mechanics [10]. This is because the probability for a state Nqb of the quantum system, or more properly the magnitude of the observable Q to which the state Nqb corresponds, can be viewed as being conditional on the next measurement (of an operator which does

not commute with the operator Q×) as well as the preparation state of the system,

described by the appropriate density operator as in the conventional formalism. This can be viewed as the physical content of the new formalism.

As is well known, orthodox quantum mechanics requires the non-local projection of the state vector, or collapse of the wave function, upon measurement to produce a new density operator for the system according to the Lüders rule. In the case of entangled quantum systems, the new density operator has to be assumed to apply at locations which are spacelike separated from the measurement that was performed. Thus it appears that some unspecified non-local influence associated with the collapse of the wave function occurs over spacelike intervals. In the present formulation, this non-locality or contextuality manifests itself through a specific mechanism in that the apparent projection of the state vector over spacelike intervals is replaced by evolution of the density operator from the next measurement backwards in time to the preparation event over intervals which are always timelike.

1579

Since the proposed pseudo-probability measure is conditional on the next measurement, it might be thought that predictions resulting from the theory are necessarily only about counterfactual properties of the system between measurements. In the case of entangled particles, the conditional state is not necessarily counterfactual because the conditionalising event can be the measurement on one particle and the prediction can be about another of the entangled particles, the measurement of which does not form part of the conditionalising event. It is therefore necessary that the predictions from the present pseudo-probabilities, conditional on the measurement of one of a pair or set of entangled particles, give the same probabilities as orthodox quantum mechanics based on the Lüders rule. This is shown to be the case in the following and the use of the formalism is illustrated particularly in relation to entangled pairs of particles.

2. – Theory

2.1. Zero-field regions. – The new formalism is presented first for the simplest and commonly considered case of a quantum system in a field free region with no time evolution of the states. The conventional expression for the probability pa(qi) that the quantum system has the property qiis

pa(qi) 4Tr (r×aP×qi) , (2)

where P×qi is the projection operator onto the subspace spanned by the normalised eigenvector(s) Nqib with eigenvalue(s) qi (which may include a range) of the observable

Q, and r×

ais the density operator for the preparation state of the quantum system. The possession of the property qi can be taken in the counterfactual sense as well as the sense of yielding the result qion measurement of Q×, although counterfactual properties do not fall within the province of orthodox quantum mechanics [11]. For a mixed preparation state r×_a4

!

_{iwiNaibaaiN, where}

!

_{iwi4 1 and Naib are the complete set of} eigenstates of A×. For a pure preparation state r×a4 P×ai4 NaibaaiN.

The conventional expression for the conditional probability that the quantum system has the property qigiven that it has the property bjcan be written as

pa(qiNbj) 4Tr (r×bjP ×

qi) , (3)

where the density operator r×_b

j for the conditional probability is given by the Lüders rule r × bj4 P × bjr × aP×bj Tr (r×_aP×_b j) . (4)

The latter expression is also the expression for the new density operator which results from the projection onto the subspace spanned by the eigenvector(s) Nbjb as a result of the measurement of B×, and was first introduced for this purpose [4]. It can be shown that if the conditional probability measure has the normal properties, including lying in the interval [ 0 , 1 ], the density operator for the conditional probability must be that given by the Lüders rule [2].

The physical motivation for the present proposal is that the properties of a quantum system should depend symmetrically on the next measurement as well as the previous

measurement. This idea has been proposed before in the time-symmetric formulation of quantum mechanics [12] and it forms the basis of the transactional interpretation of quantum mechanics [10]. Here it is given a more explicit form which can be used to make specific predictions about the quantum system. The form is different from a proposal by Aharanov, Bergmann and Lebowitz (ABL) [13] for pre- and post-selected ensembles, which has also been used as a conditional probability for counterfac-tuals [14] but which, it has been shown [15], does not conform to orthodox quantum mechanics when used in that way.

The central assumption is quite simple: that the probability pabj(qi) that the quantum system has the property qion the measurement of Q×, depends on the density operator r×

a evolving forwards in time and the projection operator P×bj from the next relevant measurement evolving backwards in time. These operators are determined by

the measurements of A× and B× which are the most recent and next measurements of

observables which do not commute with Q×. Intervening measurements of operators

which commute with Q× would not alter the following expressions. Thus we hypothesize that

pabj(qi) 4Tr (r×abjP ×

qi) , (5)

where the pseudo-density operator r×_ab

jis given by r ×_ab j4 P × bjr × a Tr (P×bjr×a) . (6)

The term pseudo-density operator is used because, while Tr (r×

abj) 41, the usual property of density operators, r×_ab

j is not a positive operator and therefore leads to the pseudo-probability pabj(qi) which may be outside the range [ 0 , 1 ] or even complex. Thus the central expression of the proposed approach is

pabj(qi) f pa(qiNbj) 4 Tr (r× aP×qiP × bj) Tr (r× aP×bj) . (7)

The pseudo-probability pabj(qi) can be viewed as the conditional (pseudo-)probability

pa(qiNbj) that the system prepared with the density operator r×a has property qi given

that it is measured subsequently to have property bj. The conditional probability

defined in eq. (7) satisfies several important properties which are expected for a conditional probability in classical probability spaces but which do not follow with the Lüders rule.

The most significant consequence of the definition is that the probability pa(qi) of orthodox quantum mechanics can be written as the sum over each of the conditional probabilities pa(qiNbj) times the probability of the corresponding conditionalising event

pa(bj). This result is always true for classical probability spaces but is true when the Lüders rule is used for the conditional probabilities only if Q× commutes with B×. In the present approach the relationship is true in all cases, as we see first for a complete set of final states

pab(qi) 4

!

j

pa(qiNbj) pa(bj) , (8a)

1581 pab(qi) 4

!

j Tr (r× aP×qiP × bj) 4pa(qi) , (8b)

where the last result, the orthodox quantum mechanical result given in eq. (2), follows only for a complete set ]bj(.

If knowledge about the final state(s) is known, the sum is not over a complete set but over a set of final states for which

!

jP×bjcI×. The probabilities then have to be renormalised to p 8a(bj), where p 8a(bj) 4 pa(bj)

!

j pa(bj) 4 Tr (raPbj)

!

j Tr (raPbj) . (9)

Thus the final expression for the probability becomes

pab(qi) 4

!

j pa(qiNbj) p 8a(bj) 4

!

j Tr (r× aP×qiP × bj)

!

j Tr (r× aP×bj) , (10)

which applies whether the sum is over a complete set or not, although pab(qi) 4pa(qi) is

independent of ]bj( if the set is complete. Note also that the sum of the

pseudo-probabilities for a complete set of intermediate counterfactual states Nqib of Q× is

!

ipabj(qi) 4Tr (r×abj

!

iP ×

qi) 41.

It would appear that the proposed expression for pab(qi) could only refer to

counterfactual properties qi of the quantum system because, by hypothesis, the next measurement on the system is of B×, not of Q×. Even in the case of repeated measurement of the same property, the new formalism would only give information about the properties in the intervals between the measurements of A× and A× and then of A× and of

B

×, or of A× and B× and then of B× and B×, rather than of the intervening measurements themselves. The possibility or otherwise of the attribution of counterfactual properties is of interest in the interpretation of quantum mechanics but the counterfactual properties are not directly relevant to real measurements that can be carried out. The current proposal does affect real measurements as well as counterfactual properties in the case of entangled quantum systems, as shown below, in paragraph 3.1.2.

2.2. Pure preparation states. – From eq. (10), for the case of a pure preparation state r× a4 NakbaakN, pakb(qi) 4

!

j abjNakbaakNqibaqiNbjb

!

jabjNakbaakNbjb . (11)

For a complete set of final states the summations lead to the familiar expression

pak(qi) 4NaakNqib N

2

. Alternatively, when the final state is known to be Nbjb, say,

pakbj(qi) 4

aakNqibaqiNbjb aakNbjb

. (12)

This is similar to the expression for a so-called “weak measurement” of Q× between the normal measurements of A× and B× [16]. The proposal here is that the probability does not involve the measurement of Q× at all but refers to the possession of counterfactual

properties by the quantum system. These could be confirmed only by another normal measurement but it is only possible to perform the measurement without contradicting the conditions of the prediction itself in the case of entangled particles and, as mentioned, this will be discussed in paragraph 3.1.2.

2.3. Expectation values. – The expectation value aQ×babj of an observable Q with

eigenvalues ]qi(, given the above preparation and measurement states, can be

calculated from aQ×babj4

!

i qipabj(qi) 4

!

i qiTr (r×aP × qiP × bj) Tr (r× aP×bj) 4

!

i Tr (r× aQ× P × qiP × bj) Tr (r× aP×bj) 4 (13) 4 Tr (r× aQ× P × bj) Tr (r× aP×bj) 4

!

i wiaaiN Q×NbjbabjNaib

!

i wiNaaiNbjb N2 , where r×_a4

!

i

wiP×aihas been used. If the preparation state is a pure state Naib, aQ×baibj4 aaiN Q × Nbjb aaiNbjb . (14)

If the final state is unknown the expression must be summed over a complete set of final states, with the appropriate weightings for the probabilities of each state given the preparation state, and then aQ×ba4

!

jaQ×babjpa(bj) 4Tr (raQ×) which is the usual quantum-mechanical result, including the familiar expression aQ×b 4 aaiN Q×Naib for the pure preparation state Naib.

Note that from eq. (13) the expectation values of the measured operators aA×babj4

!

iwiai and aB×babj4 bj are real but the expectation values of other operators may be complex, that is the expectation values of operators which are not measured can take on unphysical values.

2.4. Non-zero field regions. – In the previous section it was assumed that the

quantum system was in a field-free region and therefore that there was no time evolution of the states or the operators. We now assume that the system is subject to a possibly time-varying Hamiltonian H(t), or a sequence of time-varying Hamiltonians

Hi(t), between the measurements of A× at time tA and B× at time tB. According to orthodox quantum mechanics, the initial states Nai, tA; tAb evolve forwards in time according to

Nai, t ; tAb 4U(t, tA) Nai, tA; tAb , t DtA, (15)

where U(t , tA) is a unitary operator for which there is a well-known expression [17]. The present approach requires that after the (non-unitary) measurement process, the final state propagates backwards in time as well. Thus

Nbj, t ; tBb 4U(t, tB) Nbj, tB; tBb , t EtB, (16)

where U(t , tB) 4U21(tB, t) 4U†(t , tB), the latter results following from the unitarity

requirement on the time evolution operator. The probability pa(qi, t) that a

1583 eq. (10), pab(qi, t) 4

!

j Tr (r× a(t) P×qi(t) P × bj(t) )

!

j ( Tr r× a(t) P×bj(t) ) , (17) where r × a(t) 4

!

i wiU(t , tA) Nai, tA; tAbaai, tA; tANU†(t , tA) , (18) Pbj(t) 4U(t, tB) Nbj, tB; tBbabj, tB; tBNU †_{(t , t} B) (19) and Pqi(t) 4Nqi, t ; tbaqi, t ; tN . (20)

The right-hand side of eq. (17) can be evaluated at any time, for example, for the simplest case of a pure initial state Nakb and known final state Nbjb, pakbj(qi, t) can be re-arranged to be in the form

pakbj(qi, t) 4

aak, tA; tANqi, tA; tbaqi, tA; tNbj, tA; tBb aak, tA; tANbj, tA; tBb

. (21)

This suggests the physical interpretation that all the properties are determined at the time of preparation tA but the expression can equally well be written in terms of any other time tX, with tAE tXE tB, in place of tA in eq. (21). Therefore the physical interpretation would appear to be independent of time.

3. – Results and discussion

3.1. Entangled spin singlet. – Bell’s theorem shows that it is not possible to

reconstruct quantum mechanics on a classical probability space and retain a principle of locality which requires that the result of a measurement at one locality is not

affected by the measurement performed at another locality [1]. The usual

interpretation is that the relationships of classical probability theory do not hold because of the non-local influence of the measurement on one part and of the quantum system on the other part. As a result it appears that it is not possible to assign probabilities in a consistent way to certain counterfactual properties of the quantum system. In the present proposal, most relationships for classical probabilities do hold but the probabilities are explicitly determined by the kind and outcome of the measurement on the separated parts of the system. As already mentioned, the relationships between the classical probabilities can hold without contradicting quantum mechanics because the values of the probabilities for the counterfactuals that enter into the calculation may lie outside the range [ 0 , 1 ]. We now show explicitly how this comes about by considering several examples.

another in space in the spin singlet state

Nab 4 1

k2(N1b17 N2b22 N2b17 N 1b2) , (22)

whereN6bif N z×, 6bi represents particle i with spin 6 1

2 along the z-axis. The density operator for the preparation state is r×_{a4 NabaaN.}

3.1.2. B e l l - W i g n e r i n e q u a l i t i e s . The Bell-Wigner inequalities are briefly reviewed and then it is shown explicitly why the inequalities do not follow from the new formalism. The presentation of the inequalities follows that in Bub [1]. The case considered is where the spin of each particle can be measured in any one of three directions: r×, s× and t× which make angles 2ur, 2 usand 2 ut, respectively, with the z-axis and lie in the (x , z)-plane. The probability p(r1

1, s

2

1) is the probability that spin 1 is up along the r×-direction and that spin 2 is up along the s×-direction, irrespective of the components along the t×-direction. Because of the preparation in the singlet state, the components of the two spins along any one direction are always opposite and so only 23 configurations can occur with non-zero probability; the explicit indication that

p(r₁1, s₁2) means p(r₁1r₂2, s₂1s₁2) is omitted from the notation.

The Bell-Wigner inequality then follows by assuming that the set of particles with

r1

1 and s

2

1 is composed of two subsets: a subset which would have yielded t

1

1 (and

therefore t₂2_{) and a subset which would have yielded t}1

2(and therefore t 2

1), if spin 1 or 2 along t× had been measured. We therefore have from eq. (8a)

p(r1 1, s 2 1) 4p(r 1 1, s 2 1Nt 1 1) p(t 1 1) 1p(r 1 1, s 2 1Nt 1 2) p(t 1 2) , (23a) p(r11, s 2 1) 4x1y, say , (23b) p(r11, s 2 1) 4Tr (P × t1 1, t22P × r1 1, s12r × a) 1Tr (P×t1 2, t12P × r1 1, s12r × a) , (23c) p(r1 1, s 2 1) 4 1

2sin (ur2 us) sin (ur2 ut) cos (us2 ut) 1 (23d)

11

2 sin (ur2 us) sin (ut2 us) cos (ur2 ut) ,

p(r1 1, s 2 1) 4 1 2sin 2_(u r2 us) . (23e)

The expressions for the two terms involving the conditional pseudo-probabilities on the right-hand side of eq. (23a) are arbitrarily identified by x and y in eq. (23b), for convenience below. Individually they do not necessarily lie in the range [ 0 , 1 ] but the sum of the two terms gives the orthodox quantum-mechanical result, lying in the range [ 0 , 1 ], as shown in eq. (23e).

Similarly we can obtain the results

p(r11, t 2 1) 4p(r 1 1, t 2 1Ns 1 1) p(s 1 1) 1p(r 1 1, t 2 1Nt 1 2) p(s 1 2) , (24a) p(r1 1, t 2 1) 4x1z, say , (24b)

1585 p(r1 1, t 2 1) 4Tr (P × s11, s22P × r11, t12r × a) 1Tr (P×s21, s12P × r11, t12r × a) , (24c) p(r₁1, t₁2_{) 4} 1

2sin (ur2 ut) sin (ur2 us) cos (us2 ut) 1 (24d)

11

2 sin (ur2 ut) sin (us2 ut) cos (ur2 us) ,

p(r1 1, t 2 1) 4 1 2sin 2_(u r2 ut) , (24e) and p(t11, s 2 1) 4p(t 1 1, s 2 1Nr 1 1) p(r 1 1) 1p(t 1 1, s 2 1Nr 1 2) p(r 1 2) , (25a) p(t₁1, s₁2_{) 4z1y ,} (25b) p(t1 1, s 2 1) 4Tr (P × r1 1, r22P × t1 1, s12r ×_{a) 1Tr (P}×_r1 2, r12P × t1 1, s12r ×_{a) ,} (25c) p(t1 1, s 2 1) 4 1

2sin (ut2 us) sin (ut2 ur) cos (ur2 us) 1 (25d)

11

2 sin (ut2 us) sin (ur2 us) cos (ur2 ut) ,

p(t11, s 2 1) 4 1 2sin 2 (ut2 us) . (25e)

In the conventional treatment, each of the individual conditional probabilities

x , y , z is taken to be positive and thus

x 1yGx1y12z

(26)

and then the Bell-Wigner inequality follows:

p(r1 1, s 2 1) Gp(r 1 1, t 2 1) 1p(t 1 1, s 2 1) . (27)

In the present approach, the inequality does not follow because the individual conditional probabilities x , y , z are not necessarily all positive. Therefore with the present expressions for the conditional pseudo-probabilities, there is no contradiction with the orthodox quantum mechanics result. On the contrary, the latter follows from the foregoing equations, namely that p(r1

1, s 2 1) may be G or Fp(r 1 1, t 2 1) 1p(t 1 1, s 2 1)

since ( 1 O2) sin2_(u

r2 us) may be G or F (1O2) sin2(ur2 ut) 1 (1O2) sin2(ut2 us). There have been several proposals which reproduce the quantum-mechanical results for entangled pairs of spins [18], including ones which involve negative probabilities [19-21], but these apply just to the particular spin system considered or equivalent systems [21]. The present proposal is a general approach which provides a common expression applicable to all quantum-mechanical systems of which the entangled spins are just an example.

3.1.2. P r e d i c t i o n s b a s e d o n m e a s u r e m e n t o f o n e s p i n o f a n e n t a n g l e d p a i r . We can calculate the probability pa(q) 4pa(q11, 2

2

) that particle 1 has spin up in the q×-direction, the direction in the (x , z)-plane with polar angle 2 uq, irrespective of the direction of spin 2, by using the projection operator P×q1

1, 224 P × q1 17 I × 2, where I×i is the identity operator in the two-dimensional spin-space of particle i. On the basis of orthodox quantum mechanics, pa(q11, 2

2

) 4Tr (r×aP×q1

1, 22) 41O2 and similarly, pa(21, q12) 4Tr (r×aP×21, q2

1) 41O2. It is well known that these probabilities change if

one of the particles is measured and the outcome of the measurement becomes known.

Let us consider the case when particle 1 is eventually measured to have spin up along the z-axis and particle 2 to have spin up along the b×-direction, the direction in the (x , z)-plane with polar angle 2 ub. The predictions about particle one depend on the outcome of the measurement on particle 2, and vice versa. Therefore, in the classical case, new knowledge about the outcome of a measurement on one of the particles would normally appear in the calculation in the form of a change, usually a reduction, in the conditional probabilities that are taken into account in the calculation. As discussed above, orthodox quantum mechanics cannot be formulated in terms of relationships among classical conditional probabilities and the change in the probability must enter the formalism through the projection postulate, i.e. a change in the value of the probabilities themselves, including at spacelike separated locations. The projection postulate raises difficulties, particularly in relation to entangled states separated in space [22, 23]. Some of these difficulties will be mentioned now, before showing how they are avoided by the present formalism which deals with the situation in an analogous way to the classical case, i.e. by a change in the number of possible conditionalizing events included in the calculation, but not in a change in the value of any of the conditional probabilities. It should be noted that the following difficulties do not arise in orthodox quantum mechanics which is concerned with the predictions of measurement results only [11].

As far as the projection postulate is concerned, the most difficult problems arise when the particles are spacelike separated when they are measured. For some inertial observers, particle 1 appears to be measured first and then by the Lüders rule, the density operator r×_{a is projected onto r}×_{b4 r}×

a 84 Na 8 baa 8 N, where Na 8 b 4 N 1b17 N 2b24 N 1b17 (2sin uqN q×, 1b21 cos uqN q×, 2b2) . (28) We now have p_{a 8(2}1_{, q}2 1) 4sin 2_u

q, in place of pa(21, q12) 41O2, because of the change in the projection operator.

For other inertial observers, particle 2 appears to be measured first and then the density operator is projected onto r×

b4 r×a 94 Na 9 baa 9 N, where Na 9 b 4 N b×, 2b17 N b×, 1b24 (29) 4 (2sin (ub2 uq) Nq×, 1b11 cos (ub2 uq) Nq× 2b1) 7 Nb×, 1b2) , and now p_{a 9}(q1 1, 2 2 ) 4sin2_(u b2 uq), in place of pa(21, q12) 41O2.

Thus before the measurement both observers used the projection operator P×a to

calculate the probabilities relating to both particles but after their respective measurements, observer 1 uses the projection operator P×_{a 8}to calculate the properties of particle 2, at a spacelike separation, and observer 2 uses the projection operator P×_{a 9} to calculate the properties of particle 1, at a spacelike separation.

1587

These projection operators give the correct probabilities for the outcomes for what is seen as the second measurement in each case, but it is hard to reconcile the differing forms of the projected state vectors Na 8b and Na 9b. Since the density operator, or corresponding state vector or vectors, can be used to correctly predict the outcomes of experiments, it would appear that there must be a correspondence between the state vector and the relevant aspects of the physical system. If that is accepted, a change in a relevant aspect of the physical system must be reflected in a change in the state vector and, on the other hand, the state vector must change in a significant way only when the physical system changes. From this it would be concluded that the change from Nab to Na 8 b or Na 9 b must reflect a physical change but this is impossible because the necessary physical change involves the spin orientations of particles which are spacelike separated. Even if the state vectors are taken only as mathematical expressions, which do not reflect the physical situation except to the extent of making correct predictions about it, it is not satisfactory that state vectors like Na 8b and Na 9b cannot be transformed into one another by a Lorentz transformation [22]. The conclusion is that the projection of Nab should only occur as a local event which may change the density operator used for subsequent calculations involving the particle that has been measured but could not change the density operator used for subsequent calculations involving the distant particle which has not been measured.

The calculation proceeds differently with the new formalism. Firstly when the only knowledge about the system is the preparation state Nab, a complete set of any four linearly independent final states, for example Nbb 4N6b1N b×, 6b2, are possible in eq. (10). With a complete set of final states, eq. (10) always gives the same result as conventional quantum mechanics; alternatively, it can be easily checked that pab(21, q12) 41O2 in this case by summing over the four possible outcomes N6b1N b×, 6b2in eq. (10) explicitly.

If the outcome of the measurement on particle 1 now becomes known to be up in the

z

×-direction, the possible final states are confined to the two states N1b1N b×, 6b2 and now eq. (10) involves a sum over those two states only, which gives the result

pab(21, q12) 4sin

2_u

q. (30)

This is the key result which predicts that particle 2 has spin up in the q×-direction with probability sin2uqif the spin of particle 1 is known to have spin up along the z-axis. This prediction is made solely on the basis of the change in the knowledge about which of the possible outcomes are to be included in the calculation involving the conditional probabilities, as commonly occurs in a calculation involving probabilities in classical physics. The preparation state remains Nab, with projection operator P×a, throughout the calculation.

Note that this result is independent of the b×-direction assumed in the calculation for the direction of measurement of the spin of particle 2. Therefore we are free to test the prediction by choosing 2 ub4 2 uq, that is, by measuring the spin in the q×-direction. Since the result is the same as for orthodox quantum mechanics, the prediction will of course be confirmed by experiment. Thus this is a case, as anticipated in previous sections, in which a prediction, which might be expected within the new formalism to be necessarily counterfactual, can be tested by experiment without contravening the premises on which the prediction was made.

3.2. Elements of reality. – This subsection deals with the consequences of the new approach for a possible realistic description of the quantum system. The magnitudes of the quantum properties on measurement are correctly given by orthodox quantum mechanics and are reproduced by the present approach. The quest for a realistic description of the quantum magnitudes between the measurements necessarily involves statements only about counterfactuals. Since the present approach provides new expressions for the probabilities of counterfactuals which are more closely related to classical probabilities, a fresh look at the possibility of a realistic description of the quantum system on the basis of the new approach seems justified.

Firstly, the usual criterion for the “elements of reality” [24, 25] must be modified because the pseudo-probabilities of the theory do not lie in the range [ 0 , 1 ]. Thus the criterion is modified to be: a property of observable Q corresponding to the eigenvalue

qiis an element of reality if pab(qi) 41 and if the probabilities of all other eigenvalues of the observable, pab(qj) 40, jci. The latter part of the criterion is not usually necessary for conventional quantum mechanics because it follows automatically from the first if the probabilities lie in the range [ 0 , 1 ] and sum to unity. The viability and internal consistency of the new criterion is now considered for the examples of a single spin and an entangled singlet state for two spins.

3._{2.1. S i n g l e - s p i n c a s e . We consider a spin prepared in the pure state Nab 4N1b}

and subsequently measured to be with spin up in the b×-direction at polar angle 2ubwith the z-axis in the (x , z)-plane, Nbb 4cos ubN1b 1 sin ubN2b. From eq. (10) or eq. (12), the following probabilities can be calculated: pab(z1) 41, pab(z2) 40, pab(b1) 41,

pab(b2) 40, where pab(z1) is the probability that the spin is up in the z×-direction given the preparation and final states, etc. Thus the proposed criteria for the elements of

reality are satisfied for the component of the spin being up in both the z×- and

b

×-directions, that is the preparation and final measurement directions. In orthodox quantum mechanics, it is not possible for a quantum state to have a well-defined spin in two directions. Therefore it is worthwhile considering in more detail how this comes about in the new approach.

We can do this by calculating the expectation value from eq. (14) of the spin in an arbitrary direction S×2 u2 f, specified by the polar and azimuthal angles 2 u and 2 f , respectively: aS×2 u2 fb 4 ˇ 2 cos u[ 1 0 ] C ` D cos 2 u sin 2 uei2 f sin 2 ue2i2 f 2cos 2 u E ` F C ` D cos ub sin ub E ` F , (31) aS×2 u2 fb 4 ˇ

2( cos 2 u 1sin 2u tan ube

2i2 f_{) .}

(32)

The only cases for which aS×2 u2 fb 4ˇ/2 are u40 and u4ub, f 40 corresponding to the preparation and measurement directions which agrees with the conclusions from the probability calculation above. From eq. (14), we also find that aS×zb 4 (ˇ/2), aS×xb 4 (ˇ/2 ) tan uband aS×yb 42i(ˇ/2) tan ub. The expectation values in the x- and y-directions cannot be tested by measurement because they are contingent on the next measurement on the system being in the b×-direction. Therefore the unphysical values

of aS×xb for ubD p/4 and for aS×yb do not pose a conflict with any possible

1589

To see how the above results fit in with orthodox quantum mechanics, we can calculate the expectation value based on the preparation state alone by combining the above results for a complete set of final states: spin up in the b×-direction, which occurs with probability cos2_u

bon the basis of the preparation state alone, and spin down in the

b

×-direction, which occurs with probability sin2

ub. The expectation values for the latter are found to be aS×zb 4 (ˇ/2), aS×xb 42(ˇ/2) cot ub and aS×yb 4i(ˇ/2) cot ub. Thus combining these results, the expectation values are aS×zb 4 (ˇ/2) and aS×xb 4 aS×yb 40, from a knowledge of the preparation state alone, in agreement with orthodox quantum mechanics.

3.2.2. E n t a n g l e d s p i n s i n g l e t . Vaidman [14] has considered the elements of reality in the case of an entangled spin singlet and raised the question of a “product rule” for the elements of reality which might be inferred for the system. He used the expression of Aharanov, Bergmann and Lebowitz [13] for calculating the probabilities for the results of an intermediate measurement performed on a pre- and post-selected system for a spin singlet which is measured and found with spin 1 up in the x×-direction

and spin 2 up in the y×-direction. He then found, on the basis of the ABL approach,

pab(y21, 2

2_{) f prob (s}

1 y4 21 ) 4 1 and pab(21, x22, ) f prob (s2 x4 21 ) 4 1, where the equivalent notation [14] is shown. That is, the probabilities for each spin to be opposite to the measured spin of the other particle was found to be unity. However Vaidman also found that prob (s1 ys2 x4 1 ) f pab(y21, x

2

2) 1pab(y11, x

2

1) 40, from which he

concluded that prob (s1 ys2 x4 21 ) 4 1. Thus it appears that even though the operators

s

×

1 y and s×2 x commute with one another and have values equal to 1 with certainty, the value of their product is not the product of their separate eigenvalues, namely 1, but in fact the value of their product is 21 with certainty. This conclusion seems incompatible with any realistic interpretation but fortunately this conclusion about the elements of reality does not follow from the present approach.

As in paragraph 3.1.2, we take the case that the spin of particle 1 is up in the

z

×-direction and the spin 2 is up in the b×-direction. First we check the probability that each spin is in the direction of the final measurement of that spin. From eq. (7),

pa(z11, 2 2 Nz11, b 2 1) 41 and pa(z21, 2 2 Nz11, b 2 1) 40. Similarly, pa(21, b12Nz 1 1, b 2 1) 41 and pa(21, b22Nz 1 1, b 2

1) 40. This might appear to be sufficient evidence to conclude

that spin 1 is up along z×, but this would be incorrect. Since the conditional

pseudo-probabilities may be negative or complex, it is necessary to examine the probabilities of all the alternatives in the sub-space that is spanned by the eigen-vectors involved in describing the system. In this case, the sub-space is a four-dimen-sional spin space which is the tensor product space of the two-dimenfour-dimen-sional spin spaces of spin 1 and spin 2. When this is taken into account, the above conclusion about the spins is found to be correct because pa(z11, b

2 1Nz 1 1, b 2 1) 41 and pa(z11, b 2 2Nz 1 1, b 2 1) 4 pa(z21, b 2 1Nz 1 1, b 2 1) 4pa(z21, b 2 2Nz 1 1, b 2

1) 40, as will now be shown explicitly.

For subsequent use, it is worthwhile calculating the probability in the general case that spin 1 is up in the c×-direction with polar angle 2 uc and spin 2 is up in the

d

×-direction with polar angle 2ud. These are counterfactual properties in the present approach contingent on preparation in a singlet state and measurement of spin 1 up in the z×-direction and spin 2 up in the direction with polar angle 2 ub. The result from eq. (7) is pa(c11, d 2 1Nz 1 1, b 2 1) 4

cos ucsin (ud2 uc) cos (ud2 ub) sin ub

. (33)

We can now use, as basis vectors to span the sub-space, the states corresponding to spin 1 up and down along the z×-direction and spin 2 up and down along the b×-direction. The probabilities can be found from eq. (33) with 2 uc4 0 or p and 2 ud4 2 ub or 2 ub1 p: pa(z11, b 2 1Nz 1 1, b 2 1) 41, pa(z11, b 2 2Nz 1 1, b 2 1) 40, pa(z21, b 2 1Nz 1 1, b 2 1) 40 and pa(z21, b 2 2Nz 1 1, b 2

1) 40. As anticipated in the previous paragraph, the probability of one alternative is unity and the probability of the other three alternatives is zero, so the configuration with probability unity can be taken as an element of reality: spin 1 in the

z

×-direction and spin 2 in the b×-direction.

We can also calculate the probabilities that spin 1 is opposite to the measured direction spin 2 and vice versa, which is the case considered by Vaidman. The probabilities that spin 1 is down along the b×-direction, irrespective of the direction of spin 2, can be found by using the projection operator P×b1

2, 224 P

× b1

27 I

×

2and from eq. (7),

pa(b21, 2 2 Nz11, b 2 1) 41 and pa(b11, 2 2 Nz11, b 2 1) 40. Similarly, pa(21, z22Nz 1 1, b 2 1) 41 and pa(21, z12Nz 1 1, b 2

1) 40. This agrees with the result of Vaidman [14]. However we

shall see that it cannot be concluded from this that spin 1 down along the b×-direction and spin 2 down along the z×-direction are elements of reality.

When we examine the latter case, we take the c×-direction along the 6b

×-direc-tions, i.e. 2 uc4 2 ub or 2 ub1 p, respectively, and the d×-direction along the 6z×-direc-tions, i.e. 2 ud4 0 or p, respectively: in this case pa(b11, z

2 1Nz 1 1, b 2 1) 4sin 2_u b, pa(b11, z 2 2Nz 1 1, b 2 1) 4cos 2 ub, pa(b21, z 2 1Nz 1 1, b 2 1) 42cos 2 ub and pa(b21, z 2 2Nz 1 1, b 2 1) 4 cos2_u

b. We see that the condition that one of the possibilities has a probability of unity and the remainder zero is only satisfied for 2 ub4 p, i.e. the spins measured in opposite directions. This condition is equivalent to the one considered in the previous

paragraph. The case considered by Vaidman corresponds to 2 ub4 p/2 and then the

conditions for the elements of reality are not satisfied, each of the four possible spin configurations having probabilities of 61O2. However the probabilities do combine together to give the results anticipated above. That is

prob (b1_{411 )4p}a(b11, z 2 1Nz 1 1, b 2 1)1pa(b11, z 2 2Nz 1 1, b 2 1)4sin 2 ub1cos2ub41 , (34a) prob (z2 4 11 ) 4 (34b) 4 pa(b11, z 2 1Nz 1 1, b 2 1) 1pa(b21, z 2 1Nz 1 1, b 2 1) 4sin 2_u b1 cos2ub4 1 , prob (b1_z1 4 11 ) 4 (34c) 4 pa(b11, z 2 1Nz 1 1, b 2 1) 1pa(b21, z 2 2Nz 1 1, b 2 1) 4sin 2 ub2 cos2ub4 2cos 2 ub, prob (b1z1_{4 21 ) 4} (34d) 4 pa(b11, z 2 2Nz 1 1, b 2 1) 1pa(b21, z 2 1Nz 1 1, b 2 1) 4cos 2_u b1 cos2ub4 2 cos2ub. These results correspond to the results of Vaidman when 2 ub4 p/2. By examining the probabilities of the four alternatives, it is apparent that the conditions for the elements of reality are not met and that there is no violation of any product rule. The fact that

pa(b21, 2

2_{) f prob (b}1

4 21 ) 4 1 and pa(21, z22) f prob (z 2

4 21 ) 4 1 is due to combinations of individual probabilities in the four-dimensional spin space which include a negative probability and so the probabilities, although unity, do not correspond to elements of reality in this case. Therefore it is not surprising that the product of these probabilities results in a contradiction.

1591 4. – Conclusion

The approach adopted here is that quantum mechanics can be reformulated so that the relationships among the probabilities and conditional probabilities of the magnitudes of factual and counterfactual properties conform more closely to the relationships expected in a classical probability space. In order to achieve this, it is necessary to give up one of the axioms of classical probability because the operators of quantum mechanics do not commute and therefore the corresponding probability space must differ significantly from a classical probability space if all the axioms are preserved. The axiom that is given up is that probabilities must lie in the range [ 0 , 1 ] but this restriction is relaxed only in relation to the probabilities of counterfactual properties. It is necessary that the probability measure that achieves the above aim leads to the same probabilities for measurable properties as orthodox quantum mechanics and this was shown to be the case.

The new probability measure includes the density operator of the preparation state and the projection operator from the next relevant measurement. Thus the formulation is a specific proposal for a transactional or time-symmetric formulation of quantum mechanics. Reasons in support of a time-symmetric formulation of quantum mechanics have been advanced before [10, 12, 26]. It does not appear to have been realised that the time-symmetric or transactional approach could lead to non-classical probabilities involving negative, larger than unity and even complex probabilities. It has been suggested independently that probabilities outside the range [ 0 , 1 ] could be involved in quantum mechanics [8, 9, 27, 28]. It seems noteworthy that two relatively neglected

approaches to an interpretation of quantum mechanics, the time-symmetric

formulation and the negative or complex probability approach, should be related to one another in the way shown in the present work.

The idea that physical events occur as the result of a “transaction” between what would traditionally be taken to be the cause and effect is a significant departure from the usual concept of causality. It is worth noting that a significant departure from the traditional concept of causality is already made in the calculations for entangled states in conventional quantum mechanics because the calculations involve events at spacelike separations. The time ordering of events at spacelike intervals is observer dependent. Therefore, as discussed in paragraph 3.1.2, to some observers the experiment on one of the entangled particles is made first and used to predict the outcome of the measurement on the second particle, while to another observer the measurement on the second particle occurs first and is used to predict the result of the measurement on the first particle. Thus the classical concept of the time ordering of cause and effect appears to be called into question already in orthodox quantum mechanics. Here it has been suggested that the conventional cause and effect events should be put on an equal footing in determining the properties of quantum systems. It should be noted also that there is no formal problem involved in the evolution of the state vector or projection operator backwards in time. The new formalism avoids the problems of the Lorentz invariance of properties [29-33] because the evolution, backwards as well as forwards in time, is always over timelike intervals.

In summary, a specific formalism has been presented which brings together two relatively neglected approaches to the interpretation of quantum mechanics, namely the transactional approach and the approach involving what have here been called pseudo-probabilities. It has been shown that the formalism provides expressions for the conditional probability terms appearing in the Bell-Wigner inequalities which lead

to the quantum-mechanical result. It has also been shown that the formalism allows counterfactual properties to be assigned to the quantum system in some cases and that the product rule for those elements of reality applies as expected. In some cases the assigned counterfactual properties can be tested by experiment and they give the same results as orthodox quantum mechanics. It will be interesting to apply the new formalism to other problems in non-relativistic and relativistic quantum mechanics.

R E F E R E N C E S

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