ReDrinted from FouNDATloNs oF PHYSICS
Vol.9, Nog. li2, February 1979
Printed in Belpium
SimultaneousMeasurementand Joint Probability
Dishibutions in Quantum Mechanics
Willem M. de Muynckrl Peter A. E. M. Janssen,l'2
and Alexander Santmanl,s
Receiued Mav
5. 1977
The problem of simultaneous measurement of incompatible obseruables in
quantum mechanics is studied on the one hand from the uiewpoint ofan e:;illrt, i,treet*lenl of quanlum mechanic-c and on !h: ttlter 4snd :;tc.t!it:. i,.:,.: .. theory
of measurement. It is argued that it is preciseiy sucit c iheory of measurement
thct should prouide a meaning to the axiomatically introduced c'oncepts,
especially to the concept of obseruable. Defining an obseruable as a class of
measurement procedures yielding a certain prescribed result for the probability
distribution of the set of talues of some quantity (to be desuibed by the set
of eigenualues of some Hermitian operator), this notion is extended to joint
probability distributions of incompatible obseruables. It is shown that such an
extension is possible on the basis of a theory of measurement, under the profiso
that in simultaneously measuring such obseruables there is a disturbance of
the measurement results of the one obseroable, caused b,,- the presence of the
measuring instrument of the other obseruable. This has as a consequence that
the joint probability distribution cannot obey the marginal distribution laws
usually imposed. This result is of great importance in exposing quantum
mechanics as an axiomatized theory, since ouerlooking it seems to prohibit
an axiomatic description of simultaneous measurement of inconpatible obsert:ables by quantum mechanics.
1 Department of Physics, Eindhoven University
of Technology, Eindhoven, The
Netherlands.
2 Presentaddress:Department of ElectricalEngineering,
Eindhoven University of rechnology, Eindhoven, The Netherlands.
3 Presentaddress:Central Organization
T.N.O., Apeldoorn.
7l
0015-9018/79l020G0071
$03,00/0 @ 1979 Plenum publishing Corporation
de Muynck, Janssen,and Santman
1)
r. INTRODUCTION AND SUMMARY
l.l.
SimultaneousMeasurementRevisited
Quantum mechanical observablescorresponding to Hermitian operators
that do not commuteare generallytermed incompatible.lt is widely accepted
that (a) the Heisenberguncertaintyrelationsand (b) the von Neumann proof
that simultaneousmeasurabilityof observableswould involve commutativity
of the correspondingoperators (Ref. I, Chapter III.3) sufliciently refute
the possibility of simultaneousmeasurementof incompatible observables'
suggest
of Bohr(2)and Heisenberg(3)
Moreover, (c) the thought-experiments
incompatible
of
lor
the
measurements
arrangements
the
measurement
that
are mutually exclusive.So. not only doesthe statisticalalgorithm
clbservables
of quantum rnechanicsseem to be defying simultaneousmeasurementof
to do so.
but even experimentitself seen.rs
incompatibleobservables,
They
The problem has been raised again by Park and Margenau.(a'5)
noticethat (a) the uncertaintyrelationshave nothing to do with sirntlltaneous
sincetheserelationsregardstandarddeviationsof measurement
measurement,
results which are obtained by measuring the observablesseparately(see
The interest of Park and Margenau fcr simultaneous
also Ballentine(u)).
measurementhas been triggeredby the fact that they think, contrary to (c)'
to have found experimentalprocedureslbr the simultaneousmeasurement
Sincethe von Neumann proof (b), in the opinion
of incompatibleobservables.
of Park and Margenau,would prevent a quantum mechanicaldescriptionof
these experiments,they change the postulateson which von Neumann's
treatment is based,in such a way that the proof is no longer possibie.By
doing so they seem to create the necessaryroom to incorporate simultaneous measurementof incompatible observablesinto the theory. Their
solution consistsin weakeningthe postulatewhich assertsa correspondence
and Hermitian operators: to a certainkind of observables
betweenobservables
that could play a role in simultaneousmeasurement(the so-calledcompound
observables,cf. Section3.2) no Hermitian operator would have to be attributed.
Although we agreewith Park and Margenauthat the theory of quantum
mechanicsdoesnot prohibit a priori simultaneousmeasurementof incompatible observables,or at least that it is not certain that quantum mechanics
cannot be reformulatedin sucha way that all objectionscan be dealt *'ith, in
Sections2 and 3 we shall discussextensivelywhy we think that their solution
is incorrect.To summarize,our criticism boils down to the remark that Park
and Margenau try to reducethe problem to that of singlemeasurements.They
do this theoreticallyby means of their definition of simultaneousmeasurement, as well as experirnentallyin their choice of measurementprocedures
in QuantumMechanics 73
andJoint ProbabilityDistributions
Measurement
Simultarteous
that are to be interpreted as simultaneousmeasurements.This reduction
showsup most clearly where it is nol required that a joint probability distribution (jpd) for the two observablescan be calculatedfrom the theory. Only
the rnarginal distributions of both observablesseparately are required and
theseshould be equal to the singly measuredquantum mechanicalprobability
distributions.When one takesthe line that the statisticalalgorithm of quantum
mechanicsdoes not have to produce ajpd, one can fiope indeedthat by thus
weakening the definition of simultaneousmeasurement,impediments to such
measurementswill disappear.
However, regarding the theory of Park and Margenau, two problems
loom. In the first place the question may be raised whether measurement
procedurescharacterizedin this way by marginal distributions only, are to be
interpreted as simultaneousmeasurementsin a meaningful way. In Appendix
B we shall analyzethe experimentsproposed by Park and Margenau and we
shall reach the conclusion that the answer should be negative.
In the secondplace there is the problem regardingthe jpd. The solution
offered by Park and Margenau, viz. not attributing an operator to certain
observables,has a consequencethat the relation between the experimental
jpd and the statistical algorithm of quantum mechanics is broken. With a
view to the analogous procedure for single measurementsthis implies that
one is depriving oneself of the means to designate some functional of the
state function (or density operator) as a jpd of the incompatible observables.
This may be the reason that the jpd is treated by Park and Margenau in an ad
hoc and consequentlyvery unsatisfactory way (Ref. 4, pp.2l2tr).
1.2. On Joint Probabilitv Distributions
In discussionson the physicalinterpretation of functionalsof the state
function if representingjoint probability distributions of 1wo quantum
mechanical quantities in essencetwo views are encountered.
The first one considers the jpd in the senseof a classical probability
distribution of two random variables. The jpd of two magnitudes sr' and #
is then interpreted as the probability that .r/ and .4 harc certain values. The
magnitude, having a specificvalue, is then consideredas a property possessed
by the object system. This view is advocated,for instance, by Mayants
et al.t7tand, in a less explicit way, by Ballentine.(6)
ln Section 2.4 we shall
discussthesejpd's which we shall denote as jpd's of they'rsr kind.
The second one considers the jpd as a distribution of measurement
results obtained from a joint measurement of the quantum mechanical
observables(for instance,Jauch(8)).
This view is based,maybe provisionally,
on a more operational conception of quantum mechanicalquantities, the
74
de Muynck,Janssen,andSantman
first view presupposinga more realistic one. This secondkind ofjpd will be
discussedin Section2.5.
Becauseof the physical interpretations, stronger requirements are to be
imposed on jpd's of the first kind then on jpd's of the secondkind. In Section
2 . 4 w e s h a l lp r o v et h a t a r e q u i r e m e n ta. s s u m e db y B u b , ( el)e a d st o t h e i m p o s sibility of jpd's of the first kind of incompatible observables.
Bub's requirement is not imposed on jpd's of the second kind. These
should only obey the weaker requirements displayed in Section 2.1. ln
Sections 2.2 and 2.3, subsequently,two special classesof functionals are
discussedto see whether these could serve as jpd's of the second kind. For
both classes,of which a functional proposed by Jaucft{toris doubtless the
most plausible one, the answer is negativefor incompatible observables.Yet,
mathematical expressionsobeying the requirements for jpd's of the second
kind can be written down (Section 2.5). The physical meaning of these
expressions,however, is dubious and their relation with the measurement
processunclear.
Of course, jpd's of the second kind can be disposed of by declaring
simultaneousmeasurementof incompatible observablesto be experimentally
impossible. Yet, although we cannot propose an experimental procedure for
the simultaneousmeasurementof 9 and 9,we agreewith Park and Margenau
that the construction of a couple of examplesof mutually exclusiveexperimental arrangements does not constitute a sufficient analysisof the problem.
Also, the possibility or impossibility of certain experimental procedures
cannot be decided formally. Yet, quantum mechanicscould impose restrictions on the processesto be describedby it. However, in the caseof measurement processes,these possible restrictions can only be found by studying
such processesas quantummechanicalprocesses,describedby a Schrodinger
equation involving both the object system and the measuring instrument.
Thus, conclusionsconcerning the possibility or impossibility of simultaneous
measurementof incompatible observablescan only be drawn from a quantum
mechanical theory of measurement.In this framework the Bohrian dogma
that the measuring apparatus ought to be described by classical mechanics
is to be understoodin such a way that in principle it should.bepossible-at
least for the macroscopic part of the measurementprocedure-to perform
the classicallimit without changing the measurementresults. Doubtless this
requirement is correct, but it does not lead us much further. For, that part
of the measurementprocessin which the information is transferred from the
object system to the measuring apparatus is certainly a microscopic process
and consequently should be described by quantum mechanics. Only after
amplification is a classicaldescriptionfeasible(this point has been stressed
by Blokhintsev(11)).
The analysis of the measurement process as a quantum mechanical
$i6ulrqneousMeasurement
andJoint ProbabilityDistributionsin QuantumMechanics 7S
process will show the necessity to consider still a third kind of jpd that
are not functionals of the initial state function I of the object system alone,
but that may also depend on the measurementarrangement. Thesejpd's of
the third kind obey still weaker requirements than jpd's of the second kind,
viz. they need not obey the marginal distribution laws (l l) of Section 2.1.ln
Section 3.4 the possibility is considered of incorporating such jpd's in an
axiomatic treatment of quantum mechanics.
1.3. Quantum Mechanical Measurement Processes
we shall now formulate the requirements whicli are necessaryin order
that a quantum mechanical process may be considered as a measurement
process(l2)of, say, the observable .{. Let A be the Hermitian operator
corresponding to the observable .{ and let a* and a- be eigenvalues and
eigenfunctions of l, respectively.For simplicity we shall, unless otherwise
stated, supposethe spectrum of A to be discrete and nondegenerate.Thus
Au^:
Q^u*
(l)
A measuringinstrument for the observable.{ (.{-meter) ought to have
at its disposal, apart from a neutral "pointer position" corresponding to the
initial state x0, macroscopically distinguishable "pointer positions" corresponding to states 0^ , in such a way that finding after interaction with the
object system the "pointer position" 0* may be interpreted as the measurement result a,n. For this to be the case the only necessaryrequirement is
that the measurementprocess bring about a correlation between the initial
state ry'of the object system and the final state of the measuring instrument.
This correlation should be such that the experimental frequency of finding
measurementresult a- equals the quantum theoretical frequency i("_ i d)lr.
This requirement, usually called the measurementpostulate, is met by the
measurementscheme
| -
$
Tsy' -
Xn-
L(t^
,lt ,lt^ A 0^
(2)
which describesthe transition from the initial state x
*
xo of the combined
system of object system and measuring instrument into the final state
2^(.o^ | ,lt^>,lt* @ 0- by means of a unitary transformation T.a induced by
a suitable Hamilton operator. Here the normalized function
/,, describesthe
state of the object system corresponding to the measured valle a*. It is not
at all necessarythat ry'-equalsthe eigenfunction a- (when this is the caseone
generally speaks about an ideal measurementor a measurementof the first
kind).
de Muynck, Janssen,and Santman
76
It is sometimesfelt that the nondiagonal terms in the density operator
corresponding to ihe final state of the measurementscheme(2) prohibit an
F-or this reason Daneri g1 ql.trzt have proposed
objective interpretation.(8'12)
to endow the measurement transformation with an ergodicity property
which is sufficient "to wipe out so much of the structural details of the initial
state of the total system" that in the final state the nondiagonal terms disappear. When this ergodicity property is assumed,the measurementscheme
can be written down according to
',
Q)P^
i / 8 xo )(/o xo J9*2" o^,1,) ' I,1,,,><,1,^|
( 3)
n
in which
P*:i0,,><0*l
(4)
is the projection operator of the subspacespanned by 0,,.
Although possibly the scheme(3) is more realistic than (2), becausein
(3) it is exhibited that the macroscopiccharacter of the measuringinstrument
might disturb the coherenceof the microscopic process(2),an interpretation
of quantum mechanics as describing ensembles of identically prepared
we do not want to enter into
microsystemsadmits both schemes.(la)Since
this problem here, we shall consider both schemesin the following and find
for both similar results.
In Section 4 the measurement schemes are generalized to describe
simultaneousmeasurementsof incompatible observables.In order to provide
us with a joint probability distribution, the measurementprocedure should
yield for each microsystem a value for each observable.So we seem unable
to escapefrom the necessityof having the object systeminteracting with /u'o
measuring instruments, one for each observable. For this reason we shall
henceforth term such proceduresTblnlmeasurementsinstead of simultaneous
measurements,thus stressingthe relative unimportance of the moment of
measurement [in this sensealso the joint measurement of the observables
correspondingto the Heisenbergoperators 9(tr) and 9(tr) is a "simultaneous"
measurement.l.We propose the following definition:
A joint measurement is a measurement of two quantum mechanical
observables,involving two measuring instruments, one for each observable,
both interacting with the object system in such a way that each microsystem
of a quantum mechanicalensembleyields a value for eachof both observables.
Park and Margenau,(a'5)too, consider the possibility as indicated above.
However, they dispose of it becausethey supposethe schemesto be inconsistent when incompatible observablesare involved. In this paper we shall
question this supposed inconsistencythat stems from certain requirements
ascribed by Park and Margenau to the joint measurementscheme.
Simultaneous
Measurement
andJoint ProbabilityDistributions
in QuantumMechanics 17
In generalizingthe schemes(2) and (3) to joint measurementof incompatible observables.{ and lJ, we are confronted with the problem of which way
the frequency requirement of the single measurement scheme could be
generalized.Up until now the common statisticalalgorithm of quantum
mechanicshas not provided us with an obviousjoint probability distribution.
This circumstancemakes it impossible to require that a prescribedjpd
should be reproduced by the measurementprocedure. Park and Margenau
try to circumventthis dilemma by merely requiring that the singly measured
quantum mechanicalprobability distributions of the two observablesare
reproduced by the joint measurementprocedures.Although this requirement
seemsto be relatively harmlessand in fact is encounteredin most treatments
ofjoint measurementsin the form of the marginal distribution laws (11), it
will follow from Sections2.4 and 4. I that theselaws are only obeyedif "q/ and
B arc compatible.Then, instead of rejecting the joint measurementschemeas
inconsistent,it is possibleto arrive at an alternativeinterpretation by dropping
the marginaldistributionlaws in the form (l l). In doing so, thejoint measurement scheme boils down to a disturbance theory of joint measurement which
may be characterized by the following corollary of the proposition stated
above:
If the measurementprocesscan be describedas a quantum mechanical
process,then in a joint measurementof incompatible observablesthe measurements mutually influence each other in such a way that the singly measured
quantum mechanical probability distributions cannot be reproduced from
the measurementresults.
1.4. Discussion
The disturbancetheory of joint measurements
introducedin Section1.3
has some resemblance with the theory of measurement conjectured by
Heisenberg,13)
but actually differs from it in objective and character as well
as in results.
It was the objectiveof Heisenberg'stheory of measurementto explain
the statistical character of quantum mechanics from the "uncontrollable"
disturbance of the object system by the measuring instrument, the measurement procedurebeing describedin classicalterms: for instance,a measurement
of 9 would causean unpredictableand uncontrollable disturbanceof !4. lt ts
a bit suprising,then, that the result of the Heisenbergconjecture should be a
theory, quantum mechanics,from which the measuringinstrumentcould be
fully eliminated, whereas measurementresults could be described totally in
terms of the state function of the (isolated) object system. Such a situation
seemsto correspondbetter to Margenau'sview of the quantum mechanical
78
de Mulmck, Janssen,and Santman
state function of the object system describing the result of a preceding state
preparation.(15)
Jn this view the statisticalcharacteris an intrinsic property of
the isolated object system,describedin an objectiveway by the statefunction.
In objectivequantum mechanics(r2)
the measurementprocessis describedas a
quantum mechanical processby a Schrcidingerequation giving an objective
description of the interacting systemof object plus measuringinstrument. The
influence of the measuringinstrument on the object systemin singlemeasurements is then describedby (2) or (3), in particular by the functions f- . It is
then seenthat the measurementresults (i.e., the probability distributions and
the expectation values) do not depend ol ,lt* and consequentlyare independent of the disturbanceof the object systemby the measuringinstrument. The
possibility of viewing quantum mechanics as a theory about rneasurement
results, without explicitly considering the measurement process, is yielded
by the circumstance that quantum mechanics actually is applied to experimental procedures obeying (2) or (3). Obviously there exist experimentai
measurementprocedures(at least for singlemeasurements),the measurement
results of which are mere registrations of the initial state f of the object
system. In fact a procedure which would not yield measurementresults in
accordancewith the functionals prescribed by quantum mechanics(i.e., the
probability distributions of the values a^ of .il given by the functionals
|Q^ I ,lt)1') would not be considered to be a valid quantum mechanical
measurementprocedure.a
In Section 4 it is shown that in jointly measuring incompatible observables according to schemesgeneralizing(2) and (3), the experimental probability distribution for observable .e/ is influenced by the presence of the
$-meter, thus excluding an undisturbed registration by the d-meter (and
vice versa). So in our disturbance theory of joint measurementit is not the
disturbance of the object systemby the measuringinstrument that is essential
(as it is in Heisenberg'stheory), but it is the disturbance of the -sl-meter
by the B-meter (and vice versa) that matters. Although this picture is quite
different from that presented by the Heisenberg theory-the s{-meter not
being responsible for the quantum mechanical statistical distribution
l(F"i {,;l'of d,but on the contrary causingexperimentaldeviationfrom this
distribution-a sort of mutual exclusiveness
of measurementarrangements
for incompatibleobservablesis clearly demonstratedby it. It turns out that
certain restrictions are actually imposed by the theory of quantum mechanics
on the joint measurementof incompatibleobservables.
a We do not wantto considerherethe possibility
postulates,
of alternative
measurement
prescribing
probabilitydistributions
differingfrom ' (o- I l),, which would selecta
differentclassof quantummechanical
procedures.
measurement
In the terminology
of the presentpapersuchprocedures
wouldbe termed"disturbedprocedures."
in QuantumMechanics 79
Measurement
andJoint ProbabilityDistributions
Simultaneous
However, by theseconsiderationsit is not proved at all that it would be
impossible to construct experimental measurement arrangements yielding
for each microsystemof a quantum mechanicalensemblea value for each of
the two incompatible observables.Nor is it proved that an axiomatization
of quantum mechanicsas a theory also about joint measurementswould be
impossible.In Section 3.4 we shall discussin which way the measurement
postulatescan be altered in order to achieve such an axiomatization,if
desired.Here we mention only the main points. In the caseof singlemeasurements the theory is axiomatizedby postulatingthe functional l(cr," /)l' as
the probability distribution of the measurement results of observable .-el.
Exactly in the sameway it is possibleto axiomatizethe theory in the caseof
joint measurementsby requiring the experimentaljoint probability distribution to have some specified functional form [for instance, (24)]. Such a
specification constitutes a necessarycondition for a quantum mechanical
process to be consideredas a joint measurementprocess. A priori the
choice of the functional is largely arbitrary. However, the results of Section4
show that, when incompatible observables are involved, the marginal
distributions generated by the functional are not allowed to reproduce the
probability distributions l(."^i *'; 'z and l(p" i /)i' of single measurements
of the observablessl and 99. So the requirement of such a reproduction
should be expurgatedfrom postulatesreferring to joint measurementof
incompatibleobservables.
It seemsnatural to limit the arbitrarinesssomewhat by postulating a
weaker requirement for the joint probability distribution than the one
mentioned above, namely the requirement that for each initial state I of the
object system the correct expectation values <'lt l A i ,!'2 and <'l'l B I ,lt>
are reproduced.In Section5 it will be shown that the requirementof reproduction of the quantum mechanicalexpectationvalues guaranteesthat the
experimentalprobability distributions are such that the Heisenberguncertainty relations are satisfied, i.e.. the experimental standard deviations A.r/
and/3 fulfil the relation
Asl A:!j --, )t tlA, Bl>
(5)
Contrary to the Heisenbergtheory of measurement,which had the
precisepurpose of explaining (5), the results of quantum mechanicaljoint
measurementschemesin generalneednot meet (5): if the reproductionof the
quantum mechanicalexpectationvaluesis not required,it is very well possible
to imagine such schemesfor which the resultsviolate (5). It is remarkable
that the requirementof reproductionof the expectationvaluesimposessuch
restrictionson the mutual influenceof the measuringinstrumentsthat the
uncertainty relations are satisfiedfor the experimentalstandard deviations.
80
de Muynck, Janssen,and Santman
It seemsto be largely a matter of taste which requirements to impose
on a joint probability distribution produced by some arrangement for joint
measurement.One could even disposeof any requirement but the experimental production of a jpd. However, the piice we have to pay for this is the
impossibility of an axiomatization. The dependenceof the jpd on the measurement procedure then makes a detailed consideration of the measurement
processindispensable.
2. JOINT PROBABILITY DISTRIBUTIONS AND THEIR FUNCTIONAL
DEPENDENCE ON THE DENSITY OPERATOR
2.1. Introduction
As stated in Section 1.2, in essencetwo different physical interpretations
have been proposed for real functionals of the state function ry',which are to
representjoint probability distributions of quantum mechanical quantities.
In the first interpretation quantum mechanical quantities (which are
also indicated as magnitudesin this interpretation) are treated as classical
stochasticvariables on some probability space.In Section 2.4 a requirement,
advanced by Buf,{sr will be discussed,which this interpretation imposes on
jpd's of two magnitudes. Here we only give the following definition of a
jpd of the first kind, which is in agreementwith Bub's requirement:
A joint probability distribution of the first kind f(a*, bn) for the two
magnitudes sl and 8-represented by the Hermitian operators ,4 (with
eigenvalues e,, and eigenfunctions a-) and B (with eigenvalues 6, and
eigenfunctions F,), respectively-is a functional of the state function ry',
o b e y i n gt h e f o l l o w i n gr e q u i r e m e n t s :
positivity:
normalizatien:
I
f(a^ , b,) )-. 0
(6)
f(a",,b"):1
(7)
m,n
consistency:
/ ' ( a -, b " ) : l ( a - t , l ' > : ' | < p "oi^ ) t '
1 )l'l3*J F)l'
. f ( e , ,b, n ): : ( . p " S
(8a)
(8b)
In the second interpretation quantum mechanicsis viewed as a theory
about measurementresults. This will be discussedin Section 2.5 in more
detail. Quantum mechanical quantities are now indicated as obseruables
becausethey only are involved in the theory as far as they are encountered
in an act of observation. Observablesare, for instance,introduced by way of
yes-no experiments(Jauchtst.;.
This interpretation only dealswith operational
Simultaneous
Measurement
andJoint ProbabilityDistributions
in QuantumMechanics 81
concepts and refrains from explicit referencesto classicalconcepts such as
stochasticvariables.For this reason(seealso Section2.5) in a delinition of
jpd's of the second kind (which are pertinent to this interpretation) the
consistency requirement (8) is not compulsory. We shall replace it by a
requirement which is generally imposed on jpd's, viz. that the marginal
distributions to be calculated from the jpd of two observablesshourd equal
the singly measuredquantum mechanicalprobability distributions of the
two observables.So:
A joint probability distribution of rhe second kind p(a* , b,) for the two
observables,sy' and 3 is a functional of the state function ,/ obeyins the
following requirements
:
positivity:
P(a,,,,b))O
(e)
P l a n , ,b ) - ' I
(10)
normalization:
I
marginal distribution laws:
\ p(q^, b,) :
(o,,
,l'>:'
L p@^, bn): l< p"1r />'
(11a)
(l rb)
From the different interpretations of quantum mechanics referred to
above, it follows that jpd's of the first and the second kinds are completely
different formal entities. Yet, since functionals which obey (8) also obey
(ll), mathematicalexpressionsrepresentingjpd's of the first kind constitute a
subset of those representingjpd's of the second kind. This implies that a
functional which does not obey (9)-(l 1), and consequentlydoes not represenr
a jpd of the second kind, cannot be interpreted as a jpd of the first kind
either.
In the next two sectionswe shall discusstwo specialclassesof functionals
to be encountered in literature as intended jpd's. Both classeswill turn out
not to obey the requirementsfor jpd's of the secondkind, thus disqualifying
them also as jpd's of the first kind.
2.2. Genera[zed wigner Distributions as Joint probability Distributions
The problem of finding a functional of the state function that can play
the role of a jpd obeying relations(9) (l l) is alreadyan old one. Ever since
wigneltr.t succeededin constructing the real function that is known as the
wigner distribution, the usefulnessas a jpd of the magnitudes position g
and momenttm I of this function and of suitable generalizations(r?)of it
20)
has been discussed.(18
82
de Muynck, Janssen,and Santman
Such discussions of (generalized) Wigner distributions are generally
intended to investigatethe possibility of a phase-spacedescription of quantum
mechanics as a classical stochastic theory. For a particle (without spin) '2
and I are then consideredas the basic magnitudes of which all other magnitudes are function-s.In the stochastic theory magnitudes are described by
random variableson phasespace.Sincethe expectationvalue ofany quantum
mechanicalmagnitude is computable from the (generalized)Wigner distributhe quantum
function completelyspecifies
tion, it is seenthat this phase-space
state.
Although in this work we are not interested in a complete specification
of the quantum state by a phase-spacefunction, and we accordingly do not
require that our jpd's (especiallythose of I and 9) give such a complete
specification,it is interesting to seewhether the generalizedWigner distributions obey the mathematicalrequirementsto be imposed on jpd's. Becauseof
the context of classicalstochastictheories,it is most natural to consider them
as candidatesfor jpd's of the first kind. Seen as mere mathematical expressions, however, they might also representjpd's of the second kind. As a
matter of fact, investigationsby Cohen,(l8)Wigner,{re)3nd Srinivas and
Wolf(z0)are focusedon relations(9)-(11) (suitably generalizedto be applicable to continuous spectra).
The generalizedWigner distributions constitute a whole class(l?)of
phase-spacefunctions/(4, pi), eachmember of the classbeing some specified
functional of the state function I of a quantum mechanical object system.
Corqespondingto each member of this class,there exists a representationof
the quantum mechanical magnitudes in which the Hermitian operator ,4 is
representedby a random variable a(q, p) on phase space. For the resulting
classof representationsCohen(l8)has shown that it is impossibleto reproduce
all quantum mechanical expectation values as phase-spaceintegrals with
phase-spacefunctions f(q,p) obeying (9)-(11), if.it is required that for any
function f'the following implication is valid for the correspondenceof operators z4 and random variables a(q, p):
-->
{.F(A) <'+ F(a(q, p))}
{A <-+ a(q, p)}
(12)
The requirement (12) closely resemblesan analogous demand imposed
on the correspondenceof operators A and quantum mechanical observables
,il, which is often encounteredin axiomatization of quantum mechanicsand
will be discussedin Section3.1, viz.
{A <-+.e} =, {F(A) <-+F(s{)}
( l3)
The latter implication, which is fully on the level of quantum mechanicsas a
theory of measurement results, essentially says that a measurement of the
Simultaneous
Measurement
andJoint ProbabilityDistributionsin QuantumMechanics E3
observable-sl is effectivelyto be consideredas a joint measurementof all the
observablesF(./), their measured value being F(a1) whenever for .{ the
value au is found, au being an eigenvalueof the operator l. Clearly, (12) is
required becauseit is supposed that the random variable a(q, p) assumes
preciselythe eigenvaluesas of ,4 (cf. Ref. 9, p. 65).
The implication (13) seems very natural and consequently is seldom
questioned.Yet, in Section3.4 we shall come to the conclusionthat, at least
in a joint measurementof two incompatible observables,(13) cannot be
maintained.However,evenif (13)would obtain, it seemsto us that, becauseof
the different physical meaningsof the notions of random variablesa(q. p) and
sl, (12) is not necessary,
observables
and in fact restrictsthe classofpcssible
stochastic theories underscoring quantum mechanics in too drastic a way.
We could imagine, for instance,stochastictheories in which the values of the
random variablescorrespond with the eigenvaluesa1 only after some kind of
averaging which does not commute with the operation ^F(').
If (12) is no longer assumed,Cohen'sproof is not applicableany more,
which leaves open again the possibility that some generalized Wigner
distribution might be usable as a jpd, Also, without requirement (12),
however, it has been proved that the generalizedWigner distributions of the
above-mentionedclass(l7)donot fulfil relations (9)-(ll) at the same time.
This is known as Wigner's theorem.(re'20)
Since the class contains all correspondences between Hermitian operators ,,{ and phase-spacefunctions
a(q, p) that are commonly used in phase-spacerepresentationsof quantum
mechanics(like Wigner-Weyl, symmetric,normal, and antinormal orderings),
we would be forced by Wigner's theorem to consider more "exotic"
correspondences in a search for functionals that could serve as jpd's
of the first (or even the second)kind. For jpd's of the first kind we will
not pursue this, becausein Section2.4 we shall discussa generalargument
forbidding first-kind jpd's of incompatible observableswithout any restriction
to special classesof functionals. In the next section we shall study quite a
different functional in order to seewhether it can serveas a jpd of the second
kind.
2.3. A Functional Proposed by Jauch
In the axiomatization of quantum theory given by Jauch(8'10)
in terms of
yes-no experiments,the propositions of the physical system are represented
by the projection operators of the subspacesof Hilbert space.Thus, if Mis the subspace spanned by the eigenfunctions of the Hermitian operator
corresponding to the eigenvalue a- (degeneracy will be allowed in this
section,although the restriction to discretespectrais maintained), the probability of finding amasa measurementresult is given by the expectationvalue of
de Mulnck, Janssen,
andSantman
84
the projection operator E(M",) of the subspaceMn. Let the eigenvaluesand
eigenspacesof a second operator B be given by b,, and 1y',, respectively,
and let E(N") be the projection operator of ly', . Then, if A and B are commuting operators, the probability p(a* , b,) that a joint measurementof the
compatible observablessl and fi yields the eigenvalueam , resp. 6,, , of the
according to
corresponding Hermitian operator l, resp. B, can be given(e'10)
p(a*, b) :
(14)
tr p E(M*A N")
in which p is the density operator of the object system and M,, n 1V, is the
intersection of subspacesM^ and Nn . By (14) the joint probability is taken
equal to the probability of a single observablehaving the intersection subspace.14^ n Nn as an eigenspace.
More generally, let M(4") be the subspaceof Hilbert space spanned
by the eigenvectorsof ,4 corresponding to the eigenvaluesao, which belong
to the Borel subset/o of the reals Rl, and analogouslyi(/D) for -8. Then for
compatible observablesthe joint probability p(A', A') that a*eAo and
b, e /b equals
p(A", /')
:
(15)
tt p E(M{/") n A/(/b))
As is well known, for compatible observables E(M(/'l
commute and
E(M(A") n N(/b)) :
and .E(N(/b))
E(M(A")) . E(N(/b))
(16)
Jauch(ro)tries to extend (15) to the caseof incompatible observables,in
which casethe projection operators E(M(/")) and E(N(/b)) do not commute.
The projection operator of the intersection M(A') n 1/(/b) is known(8'10'21)
to be given in this caseas
E(M(/.) a N(/b)) : !ytt1u@")). E(N(Ab))\"
(17)
The expression(15) combinedwith (17) is to be examinedas to its suitability
as a jpd of the secondkind if A and B do not commute. We shall investigate
this question in some detail.
First of all we consider the question whether anyhow nontrivial intersection subspacesM(/") ^ N(Ab) exist if A and B do not commute. To this
end we examine Example 1.
Example 1. The Hermitian operators A and B commute on a subspace
S of the Hilbert space.* and obey the condition that S reducesthe operators
A and B.
SimultaneousMeasurementand Joint Probability Distributionsin QuantumMechanics
85
Then it can be shown (seeAppendix A) that
s:
U M*^N,
(l 8)
It should be noted that (18) implies the existenceof a common set of eigenvectors A and B spanning the subspaceS. From equality (18) it moreover
follows that if .S i.{0}, then nontrivial intersectionsubspacesMmo N, do
exist.
Also, if the commutator of A and,B is nonzero for every element/ in
tr("f + 0), then nontrivial intersectionsubspacesmay exist; a simple example
illustratesthis statement:
Example 2.
A - a$(M)
Let
+ azE(M,) | a"E(M"),
B : bi(N)
+ bzE(N) 1. 63f(N3)
Mt and -V1being one-dimensionalsubspacesand Mi ri lfr : {0} for all i, j.
Then in general the subspacesM - Mtv N2 and ly' - ffi u i/, (seeFig. l)
will have common elements,i.e., M A N + {0}, sincetwo planes through the
origin alwayshave an intersectingline.
The examplesillustrate the existenceof nontrivial intersection subspaces
for noncommuting operators.According to Jauch(lo)there also exist non-
Fig. i. The subspaces
of Example2
de MuYnck,Janssen,andSantman
t6
trivial intersection subspacesfor the position and momentum operators p
and P. This is, however,on)y the caseif An or lo is unboundeii. Ii"J'' an'i
Ap ate both bounded, then the intersectionsubspacesM(As) n t/(lt) only
contain the zero vector.
Next we consider the question of the interpretation of (15) as a joint
probability distribution of the second kind, i.e., whether this probability
fulfils the requirements(9)-(11). Of coursethe positivity of probability (15)
is not in question,so we have only to considerthe properties(10) and (11).
The marginal distribution law can be written down as
p(A', /o) r p(./', (/tt)') : p(/a)
where(/D)' is the complementof /0. tt is well known that this law is vaiid for
commuting operators; however, for noncommuting operators it is not
alwaystrue. We illustratethis point with the help of the following examples.
E x a m p l e3 .
i, j : ++/t,
Spin i1. With (15) we get P(S":1,
S,:i)
-' 0 Vii,
I jP ( S " : i , S , : i ) - 0
whereasthe marginal distribution law requiresin general
I p(s, -- i, so -.i) : P(s" : i) + o
Example 4. The analog of Example 3 for the position operator Q and
t h e m o m e n t u mo p e r a t o rP . L e t / n n - ( n , n
t 1 ] a n d ^ / p b e b o u n d e d ;t h e n
M(/n1n /V(/"9 :
{0}
Vn
So
Y,tr p E(M(/t1^ N(/"s)) - A
( te)
and the marginal distribution law is not fulfilled, since
p E ( M ( l t ' , 1n N ( / ^ o ) ) - 0
(20)
,\..t,
whereasthis expressionshould equal tr p E(M(Ae)).
given by Park
The examplesof supposedlysimultaneousmeasurements
and Margenau(a)and discussedin Appendix B suggestthe possibility of
constructinga joint probability distribution obeying the marginal distribution laws for noncommuting operators if restrictionsare rnade upon the
density operator p. An analogouspoint of view has been taken by Ross.(22)
in QuantumMechanics 87
Distributions
andJoint Prob.ability
Measurement
Simultaneous
trndeedthis seemsto work if p is restrictedto the subspaceS of Example l,
i.e., p '- E(S) pf(S). Then it is possibleto introducethe Hermitian operators
A":, E(S) AE(S) and -8.-..- f(S) rE(S) which do commute,sincecommutativit3.of A and B on S combinedwith reductionof I and B try the subspaceS
gives
lA, , B,):
AE(S)
E(S) AE(S)zBE(S) - E(S) ,BE(S)'?
- (AB - 8,4)E(S) -- 0
there will
According to the theorem of von Neuaanrr i,'aiaGarajanits)
(t5)
joint
form
obeying
probability
of
the
distribution for A" and,B"
exist a
jpd
for
I and
interpreted
as
a
the marginal distribution laws, which can be
refer
r{"
8.,
we
have
to
and
-8. For, letting M(/'") and.lV(/b')
tr p E(M(/"') n N(/b)) :
tr ps E(M(/") n 1V(/b))
with p- ,-. .E(S)pd(S). lt then follows that for this classof operatorsA and B
a joint probability distribution of the secondkind existsif p - p., i.e., if the
statesare restrictedto the subspaceS. Note by the way that property (10)
(normalization)is easilyverified.
Although seeminglysuccessful,this restriction of the statesdors not
seemto constitutea real solution of the problem of simultaneousmeasurr'
ment, since it still concernscommuting operatorsI and B (now on a subspaceS of the Hilbert space),and henceis rathertrivial. On the other hand.the
existenceof such a subspaceS, on which A and B commute,seemsnecess,rry
in view of Example2, sincehere,for densityoperatorsp restrictedrc M n N,
t h e m a r g i n a ld i s t r i b u t i o nI a w ( l l ) i s v i o l a t e d ,i . e . ,
3
\/-
p(at, b,,) .i. p(a)
/r -l
becausp
e ( a r , b " ) - 0 f o r a l l n a n d a l l p . S o a r e s t r i c t i o no f p d o e sn o t w o r k
in this case.
We concludethat ( l5) cannot be, in general,a joint probability distribution of the second kind since the marginal distribution laws are violated.
Bell(21)brings forward another argument against a description of
measurementresultsof joint measurementof two incompatibleobservables
by propositionscorrespondingto intersectionsubspaces.He statesthat the
operator E(M(A') n 1V(./b))does not correspond to a combination of a
measurementof ,c/ and a measurementof .4,but to a measurementtotally
different from this, of which the outcome cannot be predicted by knowledge
of the measurementresults correspondingto the operators E(M(/')) and
E(N(/b)).ln Section3.3 this point will be consideredin more detail.
de Muynck, Janssen,anil Santman
2.4. Joint Probability Distributions of the First Kind
ln this section we shall discussthe consistencyrequirements(8) to be
imposed on jpd's of the first kind.
The problem of jpd's of the first kind concernsquantum mechanicsas a
theory describingisolatedobject systems.Quantum mechanicsis treatedas a
classicaltheory of stochasticprocesses
in which, analogousto classicalstatistical mechanics,each magnitudehas at each moment a well-definedthough
unknown value. The measuringinstrument is thought to register for each
microsystemthe value which the magnitude,independentof the measurement,
possesses.
Quantum mechanicalstatesthen correspondto probability distributions of the magnitudes,(7.25)
which are representedby random variables.
The presuppositionsof quantum mechanicsas a theory analogous with
classical statistical mechanics--sometimes called the "ignorance interpretation" of quantum mechanics(e.g.,Ref. 9) which are essentiallythose
underlying objectivistic hidden variable theories, can be summarized as
follows:
./1.
.{2.
Quantum mechanicsdealswith ensemblesof microsystems.
Each microsystemhas for every quantum mechanicalmagnitude
at any time a well-definedvalue.
/3.
A quantum mechanical measurementrecords for each microsystem a value for the quantum mechanical magnitude which
dependsonly on the well-definedvalue possessedby the microsystem.
./4.
A quantum mechanicalensembleis describedby a Hilbert space
function /, (or density operator p).
./5.
Measurement of the magnitude correspondin-eto the Hermitian operator ,4 yields the eigenvaluerz,,,of I with probability
ll,r,,, ,lt. '.
T h e f o r m u l a t i < > n .l y' .'y ' 5 e n c o m p a s s et h
s e p o s s i b i l i t yt h a t i t i s n o t t h e i n s t a n taneousvalue of the magnitudethat is registeredby the measurement,but
some functional of thesevalues (for instance,a time average)to which the
eigenvaluesof the Hermitian operator should be attributed.
In stochaslictheoriesalso joint probability distributions of more than
one magnitudeare interestingentities.Theseare not specifiedby the above
set of postuiates.which in this respectneedsa supplement.However,because
we do not have a guiding principleto tell us which llnctionals of f should be
taken as jpd's, the best we can do is study the requirementsto be imposed
on such functionals.
Of the requirements(6)-(8) on jpd's of the first kind of two magnitudes,
Simultaneous
Measurement
andJoint ProbabilityDistributions
in QuantumMechanics 89
the first and second ones are generally assumedand need no discussion.We
shall focus here on the consistencyrequirements (8). It is essentialfor the
argument leading to the consistencyrequirements to note that according to
the ignorance interpretation of quantum mechanics,to each microsystem of
an ensemblea definite eigenvaluea* of the operator A can be attributed as
an objective property. And this should hold for any magnitude at the same
time. This implies that, accordingto this interpretation,with respectto each
magnitude a quantum mechanical ensemble should be divisible into subensembles,each correspondingto a particular eigenvalueof the Hermitian
operator representing that magnitude. In each of these subensemblesit
should be possible to attribute the relevant eigenvalueto every microsystem
of the subensemble.In view of J1 and -/5 such a subensemble,seenas a
quantum mechanical ensemble,can only be described by the corresponding
eigenfunction of the operator.
We shall now use an argument proposed by Bub (Ref. 9, Chapter VI)
to show that the ignorance interpretation of quantum mechanicsas discussed
here leads to the existenceof jpd's of the first kind of compatibleobservables
only. The Bub argument consists of a requirement to be imposed on jpd,s
of the first kind, viz. the classicalconditional probabilities to be calculated
from thesejpd's are equated with the quantum mechanicaltransition probabilities. This requirement is imposed in order that the outcomes of the
random variable theory agreewith the quantum mechanicaloutcomes and is
for this reason indicated as a consistencyrequirement. Thus, Iet us consider
two magnitudeswith Hermitian operators A and B, respectively(A, Bhaving
eigenvaluese^ , bn and eigenfunctions a- , p, , respectively).Now suppose
that we divide our quantum mechanical ensemble (with state function l)
into subensemblesaccording to the eigenvaluesa^ of the operator l. Then
the probability that in such a subensemblethe magnitude rvith operator B
has a certain value 6, is given on the one hand by the conditionalprobability
f(a^ , b")li@", | ,l'>i, [in which/(a- , bn) is the jpd] and on rhe other hand by
the quantum mechanicaltransition probability (B, I n_)12.So
f(a,,,. b")1 "* I ,/)l' -
p" I
".)lt
(21a)
Alternatively, if we start from a subdivision of the ensembleaccording
to the eigenvalues6, of the operator -B and ask for the probability that in
such a subensemblethe magnitudewith operator Ahas a certain valuea*,
we find
,l'>)': l("- | F")i,
f(a*, b")l)(F")
(2lb)
Relations(2la) and (21b) are seento constitutethe consistencvrequirements
(8).
andSantman
de MuYnck,Janssen,
90
Bub's argument amounts to the observation that in generalthe left-hand
side of Eq. (21a)is not equal to the left-handside of Eq. (21b),although the
right-hand sides are equal. In order to support the conclusion that this
consistencyrequirement involves compatibility of the magnitudes, we add a
simple demonstration5that (21a) and (2lb) are only valid if lA' Bl:0'
If follows from Eq. (21b) that
\ffo*,
b^) - l(9, ,l'>f
(22)
Substituting into (22) the expressionfot.f(a^, 6,) from (21a), we obtain
I i ( F ,L" - ) l ' l ( " - i / ) l ' : l < P " \ ' l ' > '
to
For I : P,othis reduces
I
l(F, I "-)l' l(o- I F"")i': D,no
from which, keeping in mind that
t
l ( 9 ,l " - ) l ' - l
and
0(i("-IF,,)i <l
it follows that
if l('r- I F^)l # 0
(i) l("- i F,,)l : I
i f J Q ^ l 9 " , ' >+l 0
( i i ) F o rn I n o : ( " - | F " ) : 0
So
l ( " - 1 9 , . ) i# 0 -
l("-lF")i -E,no
However, this is only possible if the functions F,o and d.m are identical
(up to a phasefactor). Sincefor every F,o it is possibleto find an eigenfunction
o- with the above-mentionedproperty, it follows that all eigenfunctionsof B
are eigenfunctionsof A, too, Since an analogousreasoningapplies for .4 and
,B interchanged, commutativity of A and B follows from the existenceof a
c o m m o n s e to f e i g e n f u n c t i o n s .
The Bub argument is very general becauseit does not refer to a special
class of functionals as, for instance, discussed in Section 2.2. It merely
hinges on the possibility of subdividing the quantum ensemble into subensembles,to be describedby a state function. It is of course imaginable that
contrary to this assumption, the subensemblesmay not be considered as
context'
5park(Ref.5, p.223)hasgivenan alternative
proof,althoughin a totallydifferent
Simultaneous
Measurement
andJoint ProbabilityDistributions
in QuanfumMechanics 9l
quantum ensembles,in which case the Bub argument would not hold. It
seemsto us, however, that this would largely frustrate common practice in
experimentalquantum physics,quantum ensemblesbeing preparedfrequently
by selectionaccording to a value of some magnitude.
2.5. Joint Probability Distributions of the Second Kind
The problem of jpd's of the second kind concerns quantum mechanics
as a theory about measurementresults. Although the object system is ru*i
considered now as an isolated system, but as an open system interactlng
with a measuring instrument, it is yet customary to formulate quantum
mechanics as a theory giving an objective description of the object system
in which the measuring instrument does not show up explicitly (cf. Section
3.1)" ln this framework the question of a classicalrepresentationof quantum
statistics is irrelevant. No specific values of the quantities are attributed to
the microsystemsas objective properties.The theory only dealswith measurement results,for which reasonthe quantum mechanicalquantities are denoted
as observables.The postulates 91--/5 of Section 2.4 are replaced bv the
following ones:
MRl.
MR2.
MR3.
Quantum mechanicsdeals with ensemblesof microsysterns.
A quantum mechanical ensembleis describedby a Hilbert space
function I (or a density operator p).
Measurement of the observablecorresponding to the Hermitian
operator ,4 yields the eigenvaluea^with probability. ("* j {)lr.
As is stressed,for instance, by Bohr,{zt the object ,yri.rir obtains its
(objective) properties only in the context of the measurementarrangement.
contrary to the ignorance interpretation, here the probabilities only acquire
a meaning if for the observablethe existenceof a measurementprocedure is
presupposed, yielding the probabilities prescribed by the postulates. For
single measurementsthis is often taken for granted, although it is realized
that only for a small number of observablesare experimentarmeasurement
procedures operative. By the same token, in extending the postulates to
describeTblntmeasurements,
the existenceof a joint measurementprocedure
should be presupposed.Defining a joint measurementas is done in Section
1.3, it is necessary-as it was in Section 2.4-Io supplement the set of
postulates MRI-MR3 with a postulate specifying the functionals that are to
play the role of jpd's. Then, if MR3 is thought also to apply to joinl measurements, we obtain for jpd's of the secondkind of two observablesthe requirements (9)-(ll).
For joint probability distributionsof the secondkind thereis no necessity
92
andSantman
deMuynck'Janssen,
to assume consistency requirements analogous to (8). This is becausethe
probability distributions pertain to measurementresults only. As a matter of
fact, the right-hand sidesof Eqs. (8a) and (8b) refer to different experimental
situations, the right-hand side of (8a) representing the probability of the
value b, in a 7-measurementwhich is precededby an ideal r/-measurement
yielding the value a*[and vice versafor the right-hand side of (86), seealso
S e c t i o n4 . 1 1 .
As shown in Section 2.2, the equality (8) involves compatibility of ,ir'
and lld and consequently would, if imposed on jpd's of the second kind,
render also this kind of jpd impossible if sr' and I are incompatible. In the
context of quantum theory as a theory about measurementresults, however,
it is possible to consider the measurementprocedures corresponding to the
different terms of (8) as nonequivalent procedures.
Joint probability distributions of the secondkind do exist. The simplest
one is the expression(26)
,l'>',t
P(a.,, b,) - l(,r*I ,l'>i'l<p"l
(23)
which fulfils relations (9)-(11), but not (8). The jpd's given by Mayantstzzr
are examplesof the generalexpression(23). Intuitively, however, this expresas a jpd for incompatible observables,since it
sion has been rejected(26'28)
does not reflect any correlation which can be expected on account of the
incompatibility of the observablessl and s8. For this reason we do not
expect that a joint measurementprocedure in which the correlation between
would yield (23). Yet, strictly speaking,
the observables;/ and I is refr.ected
if a procedure yielding (23) were feasible, rve would have to accept it as a
joint measurementprocedure for ,{ and 9. Without a detailed cousideration
of the measurementprocess,however,it is impossibleto say anything about
the feasibility of such a procedure.
For this reason we shall study in Section 4 and 5 a couple of tentative
joint measurementschemes,describingthe interaction of the microsystem
with the two measuring instruments for sl and !4 in a way analogousto (2)
and (3). Although this descriptionof the measurcmentprocessis very schematic indeed,it reflectsthe minimai requirementsto be imposedon the quantum
mechanical measurementprocess.So the conclusionsto be drawn from this
description are felt to be very general. The main conclusion we shall arrive
at in Section 4 is that jpd's for incompatible observablesobtained from these
joint measurementprocedurescannot even obey the marginal distribution
laws (11). This implies that jpd's of the secondkind, whether they exist or
not, are not relevantto the problem of joint measurementof incompatible
observablesand have to be rejected.lnstead we shall introduce in Section 3.4
still a third kind of ipd in order to arrive at a consistentpicture of the joint
measurementof incompatibleobservables.
SimultaneousMeasurementand Joint Probability Distributions in
euanfum Mechanics 93
3. JOINT MEASUREMENTS AND THE AXIOMATIZATION
QUANTUM MECHANICS
OF
3.1. On the Axiomatization of euantum Mechanics
ln section 2 rve consideredtrvo interpretationsof quantum mechanics,
in both of which the theory is thought to give an objectivedescription
of
(ensemblesof) object systems.To be sure, in order to obtain
information
about the object systemit is necessaryto have it interactingrvith (macroa
scopic)measuringinstrument.However,the information obiained should
be
describedobjectivelyby the state function of the state of the objecr
sysrem
immediatelyprecedingthis interaction.Although this doesnot seem
to be in
full agreementwith the copenhageninterpretationof quantum
mechanics,
as given by Bohr's opinipn{rt on "the impossibiiityof any sharp disti'ction
betweeirthe behaviourof atomic objectsand tl-reinteractionwith the
measuring instrumentswhich serveto definethe conditionsunder which
the phenomena appear"' yet it is generallythought to be possibleto fbrmulatequantum
mechanicsin such a way that the measuringinstrument does
not play an
explicit role. Following von Neumann,{ri quantum mechanics
is then
formulated as an axiomatic theory about measurcmentresults.'rhis
can be
obtained by means of a set of physical axioms or postulatesin
which the
physical meaning of a number of mathematical quantities
is implicitly
defined.Park and Margenau(Ref. 4, Section2) lbrmulite thesepostulates
in a
very conciseway as follows:
PIS:
The set of physical observablesis in one-to-onecorresDondence
with the set of rinear Hermitian operators on Hirbert space
having completeorthonormal setsof eigenvectors.
rf operator A
correspondsto observabl
e ,{, thenthe operat.r F1;l) coiresponds
to observableF(,r/), where,F is a function.
P2: To every ensembleof identically prepared systemsthere
corresponds a real, linear functional of the Hermitian operators,,?(r)
such that if A < >.ry',the varueof nt(A) is thc arithmelic mean (./)
of the results of -cl-measurementsperformed on the member
systemsof the ensemble.
P3: To everytype ofclosed quantum systemthere corresponds
a linear
unitary operator r (the evolution operator)such that the temporar
developmentof the density operator p for an ensembreof
such
systemsis given by p(tr) .- 'l-(tz, tr)p(tr) Tt(t, tr).
,
Park and N{argenaushow that postulatesprS and p2 entail
the less
sophisticatedpostulatesMRr-MR3, expoundedin Section2.5,
of quantum
mechanicsas a theory about measurementresults.
de Muynck, Janssen,and Santmsn
94
Measurementresultsconsist of measuredvaluesand relative frequencies.
Sincedifferent measurementproceduresmay produce one and the sameset of
measurementresults, it is necessaryto consider equivalenceciassesof such
measurementprocedures.Then such an equivalenceclass can be labeled by
the observablethat is supposedto be measuredby these procedures.
In the formalism the observablesare representedby operators. That
operator should be attributed to the observable,the eigenvaluesand eigenfunctions of which generate the measurement results according to the
postulates.This leavesopen the question ofthe correspondenceofan operator
with a particular class of measurement procedures. Nothing is said about
the problem of how to construct an experimental measurement procedure
corresponding with a particular observable or Hermitian operator. The
answer to this question depends on the peculiarities of the measurement
procedures. lt can be given only on the basis of an underlying theory of
measurement, which describes the interaction between the macroscopic
measuring instrument and the microscopic object system more specifically.
Such a measurementtheory should describethe information transfer from the
microsystemto the macrosystem.This theory should be a quantum mechanical
theory, applied to the combined system of object plus measuringinstrument.
In Section 1.3 we reviewed measurementschemesdescribing measurement proceduresyielding resultswhich are consistentwith the postulates.The
schemes(2) and (3) are constructedso as to bring about a correlation between
the final state of the measuring instrument and the initial state of the object
system which is in accordance with the postulates. Indeed, when the
measurement postulates are applied to the final state of the measuring
instrument, given by the density operators
(2) -
(3)*;
A("*l {)({ 1"*,)(*^'1,l',,>10^><0*'l
l("- I /)L'| 0^><o^l
the postulated probability distribution l(cv- | *)i' of the object system is
reproduced by the pointer positions of the ensemble of measuring instruments. Although a realistic measurementprocessundoubtedly is much more
involved than the processesrelated to the schemes(2) or (3), rve shall be
content with them for the moment and consider the unitary transformation
accomplishing the transition T.s as the quantum mechanical description of
ttre measurementprocess.
Thus far we have seen that P2 is not contingent: it is based on the
pi:esupposedscheme (2) or (3), which are to be made explicit by a more
detailed theory of measurement.However, with'respect to P2 this theory of
measurement is left out of consideration: the theory of measurementhas
in QuantumMechanics 95
Measurement
andJoint ProbabilityDistributions
Simultaneous
been short-circuitedin the axiomatization PIS P3. Regarding only single
measurementsof observables,this does not seem to have any consequences.
A vast body of experimentalresults does fit quantum mechanicalpredictions
basedon Pl S-P3. Evidentlyit is possibleto deviseexperimentalmeasurement
procedureswhich produce results satisfyingthe postulates.We can formulate
this still in another way, viz. by asserting that the postulates constitute a
necessarycondition for a quantum mechanical processto be consideredas a
quantum mechanicalmeasurementprocess.Only those experimentalconstructions that fulfil the suppressedtheory of measurementin the way prescribed
by the postulatesare allowedas quantum mechanicalmeasuringinstruments.
So the domain of application of quantum mechanicsas a theory about
results of single measurementsis restrictedby both the postulatesPIS-P3
theory of measurement.
and the suppressed
In generalizing quantum mechanicsto describe not only the results of
single measurementsbut also those of joint measurements,the following
question may be raised: Will it be possibleto do so in a way analogousto the
i.e., by meansof a set of postuiateslike PlS, P2,
caseof singlemeasurements,
and P3, without considering again the underlying theory of measurement?
When this question is answered affirmatively, the next question is whether
joint measurements are to be axiomatized by the same postulates that
axiomatize single measurementsor whether joint measurementsmight ask
for new postulates.
Regardingthe joint probability distribution, the question of the possibility of short-circuiting measurement theory amounts to the question of
whether it is determined only by the initial state of the object systemand not
also by the measurement procedure. Those who try to incorporate joint
measurement of incompatible observables into quantum mechanics by
extending or changing the postulatesshould tend to answer this question in
the affirmative. Of course the successof this procedure will largely depend
on the specific way in which the axioms are chosen. Thus von Neumann
(Ref. l, Chapter III.3) comes to the well-known conclusion that only
observables corresponding to commuting operators are simultaneously
measurable.Park and Margenau,(a)analyzing von Neumann's result in a way
to be discussedin the next section, are forced by their conviction of the
feasibility of joint measurementof incompatible observablesto change the
axioms PlS. P2. P3 on which von Neumann'streatmentis basedin order to
escapefrom his conclusions. In fact they consider it sufficient to relax the
postulateP1S into a weaker version:
strong correspondence
(Some) linear Hermitian operators on Hilbert space which have
complete orthonormal sets of eigenvectorscorrespond to physical
observables. If A <+ "{, then F(A) ++ F(.{).
andSantman
de Nluynck,Janssen,
96
postulateis intendedto generatethe
This weakeningof the correspondence
latitude of not attributing an operator to certain observables,viz. the
compound observables,which according to von Neumann as weil as Park
and Margenau should play an important role in the description of joint
measurements.In the next sectionswe shall review the von Neumann and
the Park and Margenautreatmentsin more detailand investigatethe relevance
of the replacementof PIS by Pl to the solution of the problem of joint
measurements.
3.2. CompoundObservables
Von Neumann (Ref. 1, Chapter IIl.3) introduces a sum observable.
to be denotedas sy' 1t'l .In his opinion a simultaneousmeasurelnentof ./
and4 is also a measuremettof .{ -+ 9S,becausethe addition of the results
of the measurementsof .c/ and !4 givesthe value of .'/ ' l. Consequently,
he continues,the expectationvalueof s/ 1 94 ineach stater/ is the sum ofthe
expectationvalues of s/ and of 3. This holds independentof whether -ry'and
9J are statisticallyindependent, or rvhether correlations exist betrveerlthem,
becausethe law
expectationvalue of the sum - sum of the expectationvalues
holds in general,von Neumann argues.
Combining this with the law for Hermitian operators
( ! i A r B ' ,* t - < . . 1 ' l .{l> + 1 , 1B' I i , / , )
he shows that the only operator correspondingto the observable s/ t, !"'t
can be the operator A ! B.
This amounts in the theory of von Neumann to trvo rules of corresnondence:
(i)
If .ey'* >l. then F(c/)<-+ F(A).
(ii)
lf .r;1<+ A, iJ <+ -8, and sy',!4 are simultaneouslymeasurable,then
.ry': !4<->ALB.
Supposing associativity for the r-nultiplicationof observables,von
Neumann next shows that .c,/ and :4 are simultaneously nteasurabieif and
only if the operators,4 and -Bcommute.
Although the proof of von Neumann is lengthy,his result can easilybe
has shown that,.ry'and.lJbelongingto a commuunderstood.Groenewold(2e)
(ii)
(i)
compatible
if and only if tl-reoperatorsI and ,B
are
ring,
and
tative
(ii)
involvessimultaneousmeasurability,
Since
ring.
to
a
commutative
belon-e
it is immediately clear that ::/ and ltj can be measured simultaneously if
in QuantumMechanics 97
Measurement
andJoint ProbabilityDistributions
Simultaneous
and only if A and .B belong to a restrictedset of Hermitian operators,viz.
for which lA, Bl:0. Before discussingthis result,rve first turn to Park and
Margenau.
Park and Margenau(a)introduce quantum mechanicsexplicitly in an
axiomatic lbrm, iis discussedin the previous section. To approach the
problem of simultaneousmeasurement,Park and Margcnau introduce the
concept of "compound observable,F(.ry',4)" as an operational quantity:
measure,y' anC7 simultaneously,substitutethe resultsq, b into the function
.7(a, b): the value./ , :V(o, b) is then the resultof the.F(.r/,1) measure.nent.
This in fact is a gcneralizationof von Neumann's observable.il +lj. To
determinewhich operator cilrrespondsto a compound observable,Park and
Margenau proceed in a way r,,irich seemsto be more physical than von
Neumann'slreatment anil coir:tquently migirt be less open to criticismson
purely mathcmaticalgrounds like tirose of Groenewold.('e)They start from
the observaticinthat r.vhenever
a simultaneousmeasurer.nent
of .ry' and .lJ is
performed,there is a joint probability distribution 1t(tt,,,
. b,. ; p). depending
on the initial statedensityoperator p of the object systemUpd of the second
kind). The operatorFthat correspondsto the compoundobservable.V(.{. :g\
then has to fulfil the condition
(C,)
l f , F + +. V ( : / , ) d ) a n d p ( a , n , b , ; p ) i s a n e x p e r i m e n t a lj o i n t
o r o b a b i l i t vd i s t r i b u t i o n t. h e n
I
p ( . a , , ,h , . :p \ . v ( a , , , . b , , ) .
rr pF
f o r e v e r yp
lil ,11
Next they forn, ulate (Ref. .1.p. 220) a sufficientcondition for a measurement procedureto be consideredas a simultaneousor joint measurement
scheme for.ry' and /tt. It is the condition that the measurementscheme
producesthe singly measuredquantum mechanicaldistributionsfor .// and
U, both mcasurements
of ./ and .';lreferrinsto the sameinstant of tirne.This
implies:
( C r ) T h e j o i n t p r o b a b i l i t yd i s t r i b u t i o np ( d u , ,h , , : p ) s h o u l d f u l f i l t h e
m a r g i n a dl i s t r i b u t i o nl a w s( l l ) .
W i t h ( C 1 ) .( C r ) ,a n d t h e o p e r a t i o n adl e f i n i t i o no f t h e c o r n p o u n do b s e r v able, Park and Margenau prove (Ref" 4, p. 235) that (ii) holds for the compound observable.cy' , 91. Next they reproduce von Neumann's theorem
concerningthe simultaneousmeasurabilityof .,./ and 4. However,in contrast
with von Neumann, they do not accept this result. Tl-reyargue that there
exist measurementproceduresfor the joint measurementof observables
correspondin_q
to noncommutingoperators.For this reasonthey createroom
for observablesthat do not correspondto operators,as discussedin Section
98
de Muynck, Janssen,and Santman
3.1. In fact compound observableslike .t/ + 9A are not supposedto correspond to an operator. On the basisof this new, weak correspondenceit is not
possibleto derivevon Neumann's theorem,sincecondition (C1)need not be
fulfilled. So simultaneousmeasurabilitvof incomnatible observablesseems
to have been saved.
3.3. Discussion of Compound Observables
ln Section3.1 we raisedthe questionof whetherit would remain possible
to short-circuitthe theory of measurementin describingjoint measurements.
This is equivalentto the question of whether or not quantum mechanics
formulated by means of a set of postulates that are based on an underlying
theory of measurementcould be generalizedfrom a theory about single
measurementsto a theory about joint measurements.
We shall discussnow
the suitability of compound observablesin this respect.
Von Neumann constructs on the level of the physical quantities /, lj a
new entity: sl -l U. Next he investigatesunder what conditions this new
entity correspondsto an operator with eigenvaluesa * b. In this u'ay he
reducesa physicalproblem to a mathematicalone. But it remainsa question
whether this reductionis a proper one. By adding two physicalquantirieson
the basis of the additivity of t-heeigenualuesof the corresponding operarors,
thesephysicalquantitiesare handledas if they were mathematicalquantities,
viz. c-numbers.The result is not a mathematicaltranslation ol a physical
problem, but a problem of Hilbert space theory, of which the physical
meaningis dubious.
As was mentionedin Section3.1, we consideran observableas denoting
a classof measurementprocedures,which is representedmathematicalir irl
a Hermitian operator. This representationhas to be based c-rna thet,i'_\.rf
measurement.However"addition of t.womeasurementresultsis an t,.peration
which takes place on the nracroscopicend of the measurementprocess.and
can hardly be seento have any relation to the microscopicpart of such rr
process, which yet seems to be the constituent mainly determining the
observable representingthe measurementprocess. Viewed in this li_eht
the addition of measurementresults does not imply the necessityol an
operator representingthis operation.Of course this applies to any mathematical operation on the eigenvalues.So this seemsto support Park and
Margenau's view that no Hermitran operator should be attributed to a
compound observable.
A remark madeby Bell(24)
in the contextof a discussionof hiddenvariable
theoriesis also relevanthere. lt is stressedby him that measurementsof the
spin componentsS,, S,, and (S, +,S,)/VZ can be performed by SternGerlach experiments with the magnetic fields oriented in three different
in QuantumMechanics 99
Measurement
andJoint ProbabilityDistributions
Simultaneous
directions, x, -y, and x i l. Notwithstanding that the third operator is
mathematically related to the other two by way of addition, measurementsof
S, and S,,in no way constitute a measurementof S' t S, . In fact the measurements correspond to totally different experimental arrangements, which
should be a general feature for incompatible observables.lt hardly seems
appropriate to attribute the operator ^S,+ 'S1to some compound observable
of the observables9,and gu. Again the mathematicaloperationof addition,
this time ofoperators,doesnot seemto have a counterparton the operational
level of measurement,
In order to be able to calculate properties that can be compared with
experimentalresults,each observablephysical quantity should be represented
mathematically.For simultaneousmeasurementthe joint probability distributions are the observablephysicai quantities, and the compound observables
should have the function, if any, of generating these from the Hilbert space
formalism. However, precisely by introducing the weak correspondence
postulate Pl, Park and Margenau are precluding a mathematical representation of the compound observables,thus rendering the idea of compound
observablesrather sterile. Thus, by doing so they give up the possibility of a
formal basis for calculating joint probability distributions. For this reason
the idea of compound observablescan and should be disposedof as physical
quantitiesrelevantto simultaneousmeasurement.In our opinion compound
observableshave been invented essentiallyto reduce the problem of simultaneous or joint measurementto the problem of a single measurementof the
compound observable. Although of course it is largely a matter of taste
which experimental procedures one wishes to designateas simultaneous or
joint measurements,in our opinion it is essentialthat a joint probability
distribution is generatedby two separatemeasuringinstruments, each corresponding to one of the two observablesmeasuredjointly. Compound observables cannot contribute to understandins such measurementprocedures.
3.4. Toward a Disturbance Theory of Joint Measurement and
Its Axiomatization
It was also questionedin Section3.1 whether the underlying theory of
measurementshould be changedin generalizingthe theory to joint measurements of incompatible observables.
There exist two reasons for considering the theory of measurement
explicitly.
ln the first place,by removing compound observablesfrom the theory
we have deprivedourselvesof a possibilityof tackling the problem axiomaticallv. Of course we could trv to devise other kinds of axiomatics, but
100
de Muynck, Janssen. and Santman
unlessbasedon a theory of measurementthis would be a rather arbitrary wav
to proceed.
T'he secondreason derivesfrom the fact that the consistencyrequirement (8) cannot hold for joint measurementsof incompatible obscrvables.
This suggeststhe possibilitythat, contrary to posrulateMR3 of Section2.5.
a joint measurementof .:r' and 4 on a systemdescribedby an eigenfunction
of I (or r9) might yield measurementresults which are difTerentfrom rhecorrespondingeigenvalue.This rvould invalidate each set of postulatesin
whiclr for the joint nteasurementof two incompatibleobservables../ and :l
the unique measurementresult a,, is required if the initial stateof the object
system is describedby the eigenfunctionu,,,ef A. This circumstancelvould
show an essentialdilrerencebetweena singlemeasurementof observable. y'
and the situation in which ./ is measuredjointly with an incompatible
observablel. Measurementtheory immediatelysuggestsan erplanation ol
this difference:the presenceof the g-meter obviously could influencethe
measurementresultsof the ./-measurement.In other words: the .ry'-measurement could be disturbed by the joint measurementof an incompatible
observable.Such a disturbanceof the .r/-measurementby the :,t-meter ot'
coursecan be studiedonly on thr'basis of a detailedtheory of measurement.
In Sections4 and 5 we shallratify the disturbancetheory of the joint measurement of incompatible observables,conjectured here, by studyin-eerplicit
measurementschemes.
If measurententresults depend on the explicit measurementarrancement, we may expectthat tiris will also be the casewith jpd's. Symbolizing
t h e m e a s u r e m e npt r o c e d u r eb y T , t , a , w e h a v e t o s t u d yj p d ' s t h a t n o t o n l v
dependon the initial state,y'of the object system,but also ctn7.4..,,1,
We shall
t e r m t h e s ej p d ' s o f t h e t h i r d k i n d a n d d e n o t et h e m b y W ( a , , , , b , ' . r i , . ' l , z l
For the measurementschemesconsiclered
in Sections4 and ,5ihe c,,n)i\r.nc,\
r e q u l r e m e n t( 8 ) w i l l t u r n o u t t o b e e q u i v a l e n tw i t h ( l l ) . S o t h e m a r g r n a l
distributicrnsof IV(a,,,.b,: rlt,7-,.1
.,oa)cannot equal the singll, meesurecl
quantum mechanical probability <iistributionsas is required by, ( I I ) for
jpd's of the secondkind. This result, seeminglythoroughly obstructins rirr
a x i o m a t i z a t i o no f t h e t h e o r y o f j o i n t m e a s u r e m e n its, c o m p l e t e l ya c c e p t l b l e
i n a d i s t u r b a n c et h e o r y o f t h e j o i n t m e a s u r e n e n t .T h e r e l a t i o n s ( l l ) .
which are the marginaldistributionlawsfor jpd's of the first and secondkinds,
are not the marginal distribution laws for jpd's of the third kind. For this
reasonthey rvill be termed from now on the nondisturbance
relations.since
they are satisfiedonly when mutual influenceof the ,o/- and ,'lrg-measurements
is absent,this being the caseonly if lA, B) - 0.
Although the mutual disturbanceof thejoint measurements
rvill certainlir
dependstrongly on the measurementprocedures,there remainsa possibility
o f a x i o m a t i z i n gq u a n t u m m e c h a n i c si,n c l u d i n - jeo i n t m e a s u r e m e n t \l '.o t h i s
in QuantumMechanics101
andJoint ProbabilityDistributions
Simultaneous
Measurement
end we could proceed in a way which is analogous to the case of single
measurements,defining those quantum mechanicalproceduresto be measurement proceduresthat meet certainrequirements.ln the caseof joint measurements we could specify some functional of ry'as the jpd of the third kind and
define only those procedures as joint measurementprocedures that experimentally yield this jpd. In this manner only those proceduresare acceptedas
joint measurementproceduresin which the mutual disturbanceis standardized
in a specifiedway. This will be made more explicit in Section5.
As to possiblejpd's of the third kind, it is clear that the functionri (23)
is not to be considered.sinceit satisfiesthe nondisturbancerelationsand (ll).
Also, the functional (15) does not seemto be a useful candidate.Although it
does not satisfy the nondisturtran,;erelations, the singular behavior exhibited
through (19) and (20) seemsto repudiate it effectively.
For the joint measurementof position and momentum an interesting
candidate for a jpd of this sort could be the smoothed Wigner distribution
given by de Bruijn and by Cartwrightt:ot
wc(q.p)
( , Rt7,'2 r'
';#
a, ao',tp'*" (q
[
'. e x p I i - n ,
[-..,
- L'4+fu'+l"n)
,@-_q'\2 _]Jp-pl'1
h
N
,r.,8t 0
(24)
which for c'B < 1 satisfies(9) and (10), but not (11).
In connection with the possibility of extending the axiomatization of
quantum mechanics to joint measurement, we still want to remark the
following. Since the postulatesPl,P2, P3 involve the nondisturbancerelations, they cannot be maintained if the joint measurementof incompatible
observablesshould also be describedby them. In this case,becauseof the
mutual disturbance, it is no longer desirabie that the rnarginal probability
distributions equal the singly measuredones.As shown by Park and }4argenau
(Ref. 4, p.224'),thesesinglymeasureddistributionsfollow ficm the postulates
becauseof the fact that a measurementr:f the obseriable .{ is considered
to be a joint measurementof sll observablesF(s/) fcr arbitrary functions d
the operator F(,4) being attributed to the otrservableF(-cl). If we take for
instance F(A) to be the projection cperator E(M*), then the mean value
postulate P2 applied to this observableyields the probability l(a- | ry')1'?
of
the measurementresult a- . So it is clear that the correspondencerule F(A) <-+
F(il) cannot be maintained, at least not for all d in order to arrive at a set of
postulateswhich is consistentwith the theory of joint measurements.
It is not clear whether the class of functions F could be restricted to a
larger class than just the one containing only the single operator F(A) : A.
In any case not all projection operators E(M^) are allowed. Also, it is not
de Muynck, Janssen,atrd Sanhnan
r02
sufficient to restrict the class of F(,4)'s to all polynomials of A, for in this
casethe mean value postulate P2 would require
m(Ah) :
Fr,*(o^
, bn i {, Ts,s)a,,u -
({', A^ t rl'>,
k '- 1,2...
(25)
However, this would mean that all moments of the probability distribution of sl are defined and by these the characteristicfunction m(exp irA'1.
But since on the one hand the probability distribution is determined completely by the characteristicfunction and on the other hand the samemornents
are generated by the distribution l("*l rlt)l', the nondisturbance relations
follow from (25).
The problem of which class of functions F(,4) remains to be attributed
to a certain measuringinstrument can be decided by the observation that the
usual, less sophisticated way of axiomatization referred to in Section 2.5
couid provide a picture which is consistentwith the theory of measurement.
It is sufficient, then, to attribute just one observable, say ,il, to the class of
measuringinstruments and consequentlyone operator l. Then the postulates
Pl and P2 could be replaced by a postulate determining the single and joint
probability distributions for single and joint measurements, respectively.
For single measurementsthis could be the postulate MR3, from which the
mean value rr(l) follows as a theorem. For joint measurements MR3
cannot apply. So a new postulate should be introduced, defining, analogous
to MR3, a jpd of the third kind, for instance (24). h might be desirable, as
shown in Section 5, to add a mean value postulate stating that, in jointly
measuring the observables,il and #, the measurementresults should yield
and (r/ | B l,t'>. It is easily
the quantum mechanical mean values <,1'I A 1,1'>
9.
I
(24)
for
and
requirements
these
meets
that
verified
So, nothing seemsto stand in the way of an axiomatization of quantunr
mechanicsincluding joint measurements.However, we do like to stresshere
that the feasibility of such an ariomatiz.ation can only be decided in the
laboratory. Moreover, because of the relative arbitrariness of choice as
regards the joini probability distribution, different sets of joint measurement
procedurescorrespond to different postulates.This might result in the existenceof different axiomatic systemsside by side,which describethe sanreset of
singlemeasurementsbut diferent, nonoverlapping setsofjoint measurements.
4. A QUANTUM MECHANICAL
THEORY OFJOINT MEASUREMENT
4.1. A Measurement Scheme for Joint Measurement
In this section we shall consider a quantum mechanical theory of joint
measurementin which an object systemS describedby a state function r/ is in
in QuantumMechanics103
Measurement
andJoint ProbabilityDistributions
Simultaneous
interaction with two measuringinstruments measuringobservabless/ and 8,
respectively.The observableswill be representedby the Hermitian operators
A and,B with eigenvaluesand eigenfunctionsgiven by
Aa* :
BF, :
a^d-,
brr!^
(26)
A similar casehas beenenvisagedby Park and Margenau(Ref. 4, p.268)
and we shall follow their notation. The ,il- and 4-meters are described by
initial states ;10and fo , whereas the pointer positions are characterizedby
orthogonal states d- and r1n, respectively, which are in one-to-one-forrespondencewith the eigenvalues
amand 6,, respectively,the spectraof A and
-B being supposedto be nondegenerateand discrete.
The evolution frorn initial to final state of the combined system of S
plus the s/- and Q-meters is treated as a generalization of (2) and is supposed
to take place according to the scheme
Yo:
* g xo g, go
:4-&-ryr:
t
smn({,Tot,s\$, @ 0^ e ,^
Q7l
in which id,,} is some complete orthonormal set of functions in the Hilbert
spaceof S and s1-"(ry',Te.s) are coefficientsdepending on the initial state ry'
of the object system and possibly on the measurement procedure T.s.a.
The joint probability distribution W(o*, b";.1'; T"a,s) for .{ and I may
then be definedas the expectationvalue ofthe projection operator P* E Q, :
| 0^><0^ | @ | ,t")(rt" I in the final state F, , thus yielding
W(e*,b"; *: Te,d :
I
I sr*n(lt, Ta.df
(28)
I
I
W ( a ^ ,b " : { ; T s , s )
(2e)
1
m,n
Analogous to the caseof single measurement,we would like to impose
certain restrictions on the schemeTs,s in order to have reasonto consider it
as a joint measurementscheme.Theserestrictionsshould be specifiedthrough
conditions on the coefficientssr*n(*, T"t.o).A first restriction is obvious.
The transformation Td.s, being a quantum mechanical process,should be
linear. Suppressingin the notation the dependenceon Ts.s, this implies
the condition
sm,(*):
\
(",| l) s,-"(""):
T
(p,l,l";s,^,(9,)
(30)
In the following we shall investigatea secondrestriction.The requirement
might be imposed that in the final state F1 the relevant measuringinstrument
should have the corresponding pointer position with certainty if S is initially
in a state described by an eigenfunction of A or B. This second restriction
andSantman
de MuYnck,Janssen,
l{X
actually repfesents the requirement discussed in Section 3.4 that the .r/measurementis not disturbed by the presenceof the ,/7-meter(andvice versa).
It is expressedin the coemcientsby the relations
st*n(a,):
3,',s1,,(.'.),
s,',,(F") -
8,rs,-"(p")
(31)
The relations (30) and (31) are easily shown to be equivalent to the single
set of equalities
(cv- tt'l) sr*n(o,,) :
st^,({) -
(.P,) {) sr^"(F)
(32)
have shown that a schemeobeying theserequirePark and Margenau(a,5)
ments is inconsistent if A and -B do not commute.They draw from this the
conclusion that joint measurement of observables with noncommuttng
operators according to this scheme is impossible. In order to analyze this
result somewhat further, we exhibit the influ€nce of (32) on the joint probability distributions (28). From (28) and (32) we find
W(a- , b"; *) :
i(a- | *>' W(a," , bn i u*)
: l < P " l{ ) l ' l l / ( a * ,b , ; F , )
(33a)
(33b)
which again may be shown to be equivalent to the following four relations:
W(a^, b"; ,l'): I l(", 1S)l' W(a*, bnI e,)
(34)
: I l G ' I Q ) l 'w ( a * , b , ; F , )
and
l((a^ , bn', dr) -
6*rW(q* , bn i d^)
W(a^ , b,; F") :
6n,W(a^ , b* ; Fn)
(35)
From (29) and (33) we find the marginal distributions
*@*, b.; *) : l("- | /)l'
(36a)
L w(o^,b,,;*) : lG" | ,/)l'
(36b)
4
Then from (35)and (36)we get
W(a^ , bn; o^) :
L V ( a * , b , ; F n ):
l ( o c ,I P " ) l '
(37)
So the joint probability distribution W(a^ , b"; *) is fully determined
by (33) and (37) provided (33a) and (33b) are equal. However, with (37) this
in QuantumMechanics105
andJoint ProbabilityDistributions
Measurement
Simultaneous
equality precisely constitutes the consistency requirement discussed in
Sections2.4 and 2.5. And, as we showed there, these can only be fulfilled if
lA, BI - 0, thus accounting for the alleged inconsistencyof the scheme(27)
if A and B do not commute.
Since only (27) and (33) are responsiblefor the breakdown of the joint
measurementschemein caseA and B do not commute, we shall scrutinizein
the following the presuppositions underlying these two assumptions. We
start with the measurementscheme(27) for a joint measurement.It is possible
to write down the scheme(27) alternatively according to one of the follc 'ving
ways:
rgn,f ,e0,,, ( 3 8 a )
(/ :, {.,t',t,, 1a\!)'
s,,,,({\4,
; [I''
ez',,t,
(,/ir x,)c, (n-1t!-'T lL s,^,(*)Qt
E 0,,,f
(38b)
So it is obvious that the joint measurementof sl and lj can be considered
either as a measurementTs(/J) of .{ on the systemconsistingof S plus the
%-meter,or as a measurementTs(il) of .4 on the systemconsistingof S plus
the .q/-meter. So the scheme (27) is consistent with both nonideal singleobservablemeasurementschemesTs/((/j) and Ts(.r/). The equality of 7"4,s ,
Tg($), and T6Q/) shows that the joint measurement should be possibie
wheneverthe single measurementsare. As long as only (21) is considered,
quantum mechanicsdoes not presentany impedimentto joint measurements
that does not also concernsinglemeasurements.
We now turn to the assumption(33), which will turn out to be lar more
liable to suspicion.This may be seen as follows. Supposewe perform an
ideal singlemeasuremenlof .{ on the systemS. Then, after this measurement
the subensembleof systemsyielding the result u,t can be describedby the
state function ct,,,. Subsequentlywe can measure -'./ and !t9 jointly on the
m i c r o s y s t e mosf a s u b e n s e m b l e , r , T
, ,h. e n t h e r i g h t - h a n ds i d eo f ( 3 3 a )m a y b e
joint
probability
distribution of ,cy'and I corresponding
interpretedas the
to this procedureas a whole. However,(33b)may be interpretedin an analogous way as the joint probability distribution of a joint measurementof .c1and
98 precededby an ideal single measurementof .1. Norv, as stipulated in
Section3.3, if IA, B] + 0, measurementsof .r/ and !4 are to be considered
as totally differentphysicalprocedures,a situation that is not altered by the
joint measurementof .d and ,14.So thereis not very much reason
subsequent
to expectthat the two differentproceduressketchedaboveuill give the same
resultsfor the joint probability distributions. lndeed, it is hardly surprising
that, as has been shown in Section2.4, equality of the two joint probability
distributions (33a) and (33b) involves commutativity of y' and ,8, thus
indicating that only in this caseare the two physical proceduresdescribed
de Muynck, Janssen, and Santman
above sufficientlysimilar to render the same results.From the foregorngue
arrive at the conclusionthat the joint measurementscheme(21) might be
possible,but that the assumption(33)is too heavya requirementif [r. B] ,' 0.
Dropping it or exchangingit lor a milder one could save the scherrre.Since
{33) directly follows from (30)-(31).this would mean relaxingone or Frothoi
j(t)
the requirementsleadingto theserelations.Now the linearity conditions(
*'hat
are beyonrl doubt. Therefbre the nondisturbanceconditions 13ii are
c a n n o t b e m a i n t a i n e di n j o i n r m e a s u r e m e not f i n c o m p a t i b l eo b ' e r r u b l c . .
as was coniectured in Section 3.4. lncompatible observablescannot Lre
ineasuredjointly accordingto the scheme(27) withgut mutua) dt-'iurbance.
4.2. The Role of the Marginal Distribution Laws
It turns out that the marginal distributions of the joint probabilitl
distribution (28) piay a very decisiverole. In order to study this, $'e return
to the measurementschemesas given in (38). From theseschemeswe get the
single probability distributions of .cl and ll in the presence cl ihe c'ther
measuring instrument as the expectation values of P,n '-',0,,,' 0,', an.l
-Q,
\,)(.Tn i in the final stateYi :
ps{a,,,, ./4)-.- I ,r,,,,,(/r)d,a ! , ; T , ) t-" | 1 s , , , , ( y ' ' ) '
1'19a)
llr
'.,
ps(b" , .r'/) ..- ) , s , , , ,,,!(t $ , , . 0 ^ ' , , - I
s7,n,,(,f)r
( - l 9 h)
tm
;,
Comparisonwith (28) showsthat. independentof any extra requrr.mrrrt
on the coemcientss,-,(l) or the joint probability distributionsW(.o,', A : " i.
the measurementscheme(27) guaranteesthe marginal distributilr ierii
V , ' ( a u, ,b " ; , l i )
( -+()it )
p.s\b,,.'/) - L W(s,,,b,,; ,l')
(-+0h)
!.ttt,,, , D) ..,\
These equalitiesare valid whether the .,a/-and l-tneasurement disturb
each other or not. The requirementof nondisturbancenow can be erpressed
through the equalities
-- :,(.a^
(.+la)
*.;',
P.qt(a,, , 14)
pa(b, , .il) -- l(,F"t,{'lt"
whichare seento coincidewith the marginaldistributionlaws(16)
(4lb)
Simultaneous
Measurement
andJoint ProbabilityDistributions
in QuantumMechanics107
We alreadyshowed that (36) or (41) follows from (33). As can be seen
from (23),the converseis not true for generalexpressicn-(
for joint probability
distributions.It is. l-lowever,simple to demonsiraterhrit-for the joint probability distributions (20), related to the measurern',:1"t1
procedure (27), the
relation (33) follorvs from (36). [:ar" ll'(a,,,, h*; tj;.1being positive semidefinite,(40) and (41) give
m*m'
l4'{a,,,.itn' ; P^} '- ()'
n * n'
t42)
which again is an expressjr-in
of the absenceof mutual disturbance.Now ibr
the joint probability disiilbutif'!ri {7-8)the validity of (31) is inferred from
(42). This. to*qether
with the iiruarit;.-condition (30) againgives(32) and (33).
Sincein inost treatmentsoi'1oint probability distributionsthe marginal
distribution laws are required in the form (36), we see that this essentially
amounts to the requirementof nondisturbanceof the measurements.
From
Section2.4 we may draw the interestingconclusionthat for the measurement
process(27) the nondisturbancerelations(41) alreadyinvolve the commutativity of A and B. So, although it is possibleto devisea measurementscheme
for the joint measurementof two observableswith noncommutingoperators,
it is not possibleto do so in a way that the measurements
do not mutually
disturb. In casejoint measurementsare definedas proceduresobeying (41),
as is done by Park and Margenau,{a'st
11't.scheme(27)isruled out if lA, B) + A,
in agreement with their conclusions.However, if we accept the possibility
that joint probability distributionsare not functionalsof statealone,but may
depend as well on the measurement procedure T"r',s, then it is not very
c o n s i s t e n t o r e q u i r e( 4 1 ) . T h e d e f i n i t i o no f j o i n t m e a s u r e m e nat s g i v e n i n
Section 1.3 seemsto be more suited, the existenceof an experimentaljoint
probability distribution being the only desideratum. The measurement
scheme (21) certainly meets this definition and consequently it can be
consideredas a valuablejoint measuremenrschemealso if' lA, Bl + O.
4.3. Some Remarks on the MeasurementArrangement
The dependence
of the joint probability distribution on the measurement
procedure is a reflection of the way in which the measurementsare mutually
disturbing.So, in order to be able to interpret the joint probability drstribution theoretically, we are led to perform a detailed analysis of the measurement procedure.As is shown by (38),information is yieldedabout the object
systemin interactionwith another measuringinstrument.
The dependence
of the joint probability distribution of two incompatible
observableson the measurementproceduremay be exempli{iedby the follow-
de Muynck, Janssen,and Santman
10t
(b)
of sprn.
components
experiments
asjoint measurements
of different
Fig. 2. Stern-Gerlach
ing rather trivial joint measurementschemes.A Stern-Gerlach experiment is
performed on spin-+ particles,measuringthe x component of the spin. The
resulting beams are subsequentlyanalyzed by other Stern-Gerlach magnets
now directed in the z-direction.Since each channel (i), i - 1,2, l, .1 lcf.
Fig. 2a), correspondsto a measurementof S, and S, on the samemember of
the ensemble,this schemeobeys the definition of a joint measurementof
S, and S,. Since each ofthe channelscorrespondingto a beam learing a
Stern Gerlach magnet can be describedby an eigenfunctionof the relerant
spin operator (when the particle mass is big enough),the joint probabilitl
distribution of this measurementprocedureis of the form (33a) with r,,, the
eigenfunctionsofS, and B, thoseofS, . Then the procedurecorrespondingto
(33b) is given by Fig. 2b. The examplevery clearly showsthe inadequacl of
the requirementthat (33a) and (33b) should be equal. Indeed, Figs. 2a and
2b seem to be the opposite extremesof a whole range of possiblejoint
measurementproceduresof S* and S, , of which, however,it is quite obscure
how theseextremescould be joined in a continuousway. The more symmetric
arrangementthat Jauch (Ref. 8, p. 75) proposes,in which the measurements
in QuantumMechanics109
andJointProbabilityDistributions
Measurement
Simultaneous
are repeated in an infinite sequence(called a filter), might be thought to
belong somewherehalfway between these extremes.
4.4. A Random Phase Scheme
To conclude this section we'vant to consider a measurementprocedure,
different from (27), in which the disturbance of phase correlation (cf. Section 1.3)has been taken into account. Generalizingthe procedures(38) to this
circumstance,we get the two single measurementschemes
.t, '.'o';,:'xo
1
I / I f'X,l sl f'
--Ts/\A)
t*
\-
L
\-
L
vxo S -xn
TB(.v)
s-
sr,,,,{g'1sf,,,n,(,1'|
, 6, @ n)(6r
:
\-
L
@ nn.t(E P*
(43a)
6n 6o
tr,,,(*)t|,,',(*)'
$,
(8 0^)<.Q,,(l) 0,,", I
Q" g3b)
describedby (a3) representa unique quantum
Since the ,.""*".-",,ons
mechanical process,the right-hand sides of (43a) and (43b) should be equal'
This results in the joint measurementscheme
@P,,,.I) Q,
l,l, 8 €rOxoxry'E 4" I xo J44- L'i/,*,(.1')><'1,,,,(,lil
(44)
$ , , . ( , l r- )L r r , , , @ ' 1 4 ,
(45)
Comparing this result with (27)' we seethat also in the joint measurement scheme(44) the phasecorrelation disturbanceremovesthe nondiagonal
parts from the final state density operator. Sincethe singleand joint probability distributions following from (44) are exactly the same as those consideredbefore, the marginai distribution laws (40) remain unchangedand the
nondisturbanceconditions (41) continue to involve (42). It is, however,not
directly possibleto derive (33) (which again would involve commutativity of
A and B) from (42), becausethe scheme(44) has lost the linearity property
obeyed by scheme (27). We could derive (33) if the ergodicity of the joint
measurementschemealso would imolv the relations
I, o-)(o^,i
8 Po @ Qo kQ-
O,
m * nt
(a6a)
1P')'.F,'
C- Po O Qo k*
o.
tr - ti
(46b)
ll0
de Muynck, Janssen,and Santman
which are direct generalizationsof the corresponding behavior in sincle
measurementschemes.Then (44) together with (42) gives
I o,, (& 6o6 xo)(", I6n I xoi
-::y:!-L
W(o,", bn, e,n) ,/1,,,("^))(,/,I"n(o,n)
€.' P-, . e)., (47)
1T
in which ry'|,,.is the normalized ,!,,,n, and
i , l ' 8 ) i l @ x o ' ; ( . . | '68o€ , x o
:=
-:.
,lt;<.$
, t,,,, t", (Ilrfo E, xo)(-,,,e 6o 1,,
Ir,,r"*
-:-s.z- I , ... S';,' W(o,,",
b,: o,,),,,lrl,r(",,))(rlrl,,,(r,,)p,,, V "
-7'
n,.o,,,,
b,; ,1,)$i,({)',,(*1,,,(,/)t
a P,,,(i Q,,
( 4 8)
Similar expressions
are obtainedif I is expandedin the -B-represenrarron
Llnicity of the schemethen requires (33) and
,J'X"t$.- $Ln(',,)- ,/y,,(p,,)
r : 1 9)
So, apart from commutativity of ,4 and B, the nondisturbancerequrrement also has as a consequence
that the channelstatesrf), are independenr
of the initial stateof the system.As before,relaxationof this nondisrurbance
requirementopens up the possibility of the joint measurementschemer.1-1
)
also if [A, B) .+ O.
5. A RESTRICTED CLASS OF.IOINT MEASUREMENT SCHI }IES
5.1. Reproductionof Averages
In Section 4 we showed that the measurementschemes(l;t snd (.1.{t
may be interpretedasjoint measurementschemesof incompatibleobserrable:
when we properly account for the mutual disturbanceof the measurements
by rejectingthe consistencyor nondisturbancerequirernent(-13).Until no*
no new restrictionshave been imposed on the measurelnentschemest r the
j o i n t p r o b a b i l i t y d i s t r i b u t i o n s r, e s u l t i n g ,a s d i s c u s s e di n S e c t i o n . l . _ 1i n
. a
wide variety of arrangementsto be acceptedas measurementarrangenrents.
Up until now the characterof the mutual disturbancehas been left outside
our considerations.Yet this does not seemto be justified. For instance.*e
could imagine a scheme(27) for all n yielding as measuremenrresult ,r,,,,
Measurement
in QuantumMechanicsll1
Simultaneous
andJoint Probabili$Distributions
when the initial state of the object system is I .- cr- . Such a scheme is
hardly acceptabie as a measurement scheme of the observable sl . What
seems to be acceptable is a -situationin which at least the au-erageof the
measurementresultsequalsa,nfor this initial state.Qualitativelythis amounts
to a specificarronof the influenceof the disturbanceby the .#-meter of the
./-measurement as a broadening of the distribution function p",tfu- , 9)
reiative to its singly measured quantum mechanical shape ir.cr,,,
i /)l'. It
procedures
thus sec'nrs
rensonableto selecta specialsubclassof measurernent
(27) as the cla:s of joint measurementproceduresof the observables.r and
.4, viz. that class for u.hieh the experimentalaveragesof the measurement
resultsfor .c/ and,l ar^er'-iualto their singly measuredquantum mechanical
v a l u e s .i . e . .
ll,'ia,,, . h, ; r!t)a,,, -. (.tlt A ,lt
(50a)
I
T
/- W(a,n, b"; Q)b" -- (..* B , tl)
il
(s0b)
ft
Except in a two-dimensionalHilbert space,the requirements(50) can be
shown to be weakerthan the conditions(41), which involved nondisturbance
and commutativity of I and 8. In a two-dimensionalHilbert spacerelations
(50) imply the nondisturbancerelations.So measurementprocedures(27) for
the joint measurementof two different componentsof spin do not seemto
exist. Indeed, neither of the examplesconsideredin Section4.3 obeys (50).
However, in generala relaxation of the requirementsimposed on the joint
probability distribution in the senseof (50) seemsto createenough room for
these schemesto be valid candidatesto play the role of joint measurement
schemesof incompatibleobservables.
5.2. HeisenbergUncertainty Relations
An interestingquestion in studying the measurementschemes(27) and
(44) is whether the measurementresults for observables.r/ and ,/J obtained
in this way obey the Heisenberguncertaintyrelations.The answeris not at all
obvious sincethe nonvalidity of (41) in tl.recasethal lA, Bl + O preventsthe
experimentalroot-mean-squaredeviations 4,il znd /!d from being computed
as the square roots of the expectation values in the state ry' of the object
systemof the operators(l .-- ,l.A))2and(B - (.8)2, respectively.
The mutual
disturbances of the .{- and .7-measurernentscausesthe experirnental rootmean-squaredeviations to be different from their singly measuredquantum
mechanical values /,il
and /0.4. Furthermore, measurementschemes(27)
and (44) seemto be general enough to allow ld and /9 to be smaller than
.Iril and /09, respectively,in which case the Heisenberg uncertainty relations would be violated.
andSantman
de Muynck,Janssen,
ll2
In this section we shall show that requirements (50) imposed on the
measurementschemes(27) and (44) are sufficient in order to have
/s/
> /0.{
ls > aos
(51)
{52)
Qualitativelythis could be expectedfrom the reasoningof Section ,5.1,
since a broadening of the distribution of the measurement results 4,, will
tend to increasethe standard deviation Asr'.
and A4 of
In order to study the root-mean-square deviations /s/
the measurementscheme(27) we define the operators
A :
on the Hilbert spaces of the ddefined in Section 4. Then
\ a^P,,,
(5 3 a )
B -Lb"Q"
(53b)
and !1&-meter,respectively, P^ and Q,, are
[A,B]-0
and
(/.cr'f
L pr@,,. 9)a^2- lL r.*ro,,. vha-]
: (Y,lG - (V,\ AIY):)'lY) : A,2(A)
(54a)
_
(/rrt, _ L po(h,. .,y'tbn2
s/\b,]'
lL ooru,"
: (.Yr (B - (.Yt B I V)), )Y) : A;(B)
(5-+b)
where P, is the final state of the measurementprocess,describedby' the righthand side of (27). Although /.{ and /t4 are now expressedin the usual
way as functionals of a Hilbert spacevector, we cannot draw conclusions
directly from (54) regarding the product /s/ " Ag4 becauseA and B commute.
Restrictionson this product should result entirely from restrictionson the
possible final statesY7 . Writing
Vt :
UYi
whereY; is the initial stateof the measurementprocess(27) and Li the unitarl'
operator generatingthe transition Ts.s, we see thatY, is not arbitrary for
two reasons.In the first placeVr, being a product function, is not arbitrary,
and second,the operator U should representa measurementschemeobe-ving
(50).
in QuantumMechanics113
Simultaneous
Measurement
andJoint ProbabilityDistributions
Using I/, we can expressthe root-mean-squaredeviations in terms of the
initial state 9, of the measurementprocessaccording to
/s,/ :
ag :
/i(l-rAu),
llu-lBu)
The operators U lAU and LI lBU commute on the direct product
Hilbert space of object -1,{-meter *94-meter. So they have a common set
of eigenfunctionsin this space.The eigenvaluesbeing the same as those of
A and -8, we may parametrize the eigenfunctions according to ,lL*,,. (a
d e s c r i b e sd e g e n e r a c yr e l a t i v e t o t h e A a J B s p e c t r u m ) .T h e n { , y ' . , . , i s a
complete orthonormal set of functions in the above-mentionedHilbert space.
Since (J,!^n"is a comrnon eigenfunctiot of A and B (with eigenvaluesaand bn, respectively),we have
LIL d,nn.{^no :
(55)
,ltu,n(ld,,n"})0n,rl,
Here d^noare coefficientsand ,lt^,({d,,,*i) is a state function of the object
system which depends on these coefficients.Developing the initial state P1
according to
Y,:{
8xo86o:
dkno.p,n,o
I
:";
A"l,tk^r'
(s6)
(s7)
we can write the measurementscheme(27) as
Yr - UYo: L,!^"({akn^\)0^\,
(58)
(28)is givenby
Thenthejoint probabilitydistribution
:
ly(a^,b,; {: Te d : l\,1'^,({dk."})ll'
l dk^"\'
4
from whichA.& andA9 followas
(/.il)':
ug)'
(59)
' ^\"
i d k , " l 'e * ' - ( 2 | a h * , lo
(60a)
mnd
\mnd
I
d ' ^ n " l ' b ,-' ( L , d k , , t ' , b , l
\m-n d
|
(60b)
I
mnd
I
Incidentally,we notethat if the function2o d*norlt^nn
of (55)is a possible
process,we would have/s/ : Ag : 0.
initial stateYo of the measurement
(56)to
We now specialize
e*oEXoOfo:
I
Snto.l,
umnoYmnn
(61)
de Muynck, Janssen,and Santman
tt4
which, becauseof the orthogonalityof the left-hand side state lunctlorl: frrF
differentvaluesof mr, imPlies
\/t
,mo* trro'
-" nt na" rnna
A
"m;nn
tt'lt
Then
dkn, - | (o,noi .1, dXX"
(6-lI
ffio
which may be subsitutedinto (60a).
Disturbance of the "e/-measurementby the Z-meastrement riould be
absent if an initial state rl : d-o would yield the measurementresult 4..,0
with certainty for each lrlo . This would be the caseif for all nt and //r,,
d:' ^-' i:i,":6- ^dff5,
( 6-1.r
)
Then the joint probability distribution (59) reducesto
W(o^, b, : ,l';Ts.o) -- l("- I /)l 'L I d#,,"1,'
from which, in conjunction with (62), the equality (41a) follows. Alstr. J :,/
would reduce to /os/.
An analogous reasoning can be given for the #'measurement, lc-.iding
to the nondisturbance conditions for all n and no
d#," :
il:n" -
6nnodlnonoo
( 6.1b,
)
in which dfion,arethe expansioncoefficientswhen in (61) a-" is replacedbr
"
In Section4we showedthat if lA, B!+ 0, mutual disturbancei: c::eDtially present, so (64a) and (64b) are not aliowed simultaneouslr. Thr;
means that at least for one &m0or p,o the measurementresult can be difl'erent
lrom the correspondingvalue Qmoorbno. In general"realistic" measurement
procedures of course we expect that disturbance will take place for most
a ^-o a n dp n o .
In ord-erto show that (51) and (52) hold, we express(50a) accordingto
I i I 1 * * , 1 * ) # l io" l^' : l l ( o * l S ) la' ^
mnd'
m
m^
Sincethis shouldbe valid for arbitrary ty',it is equivalentto
T
c l ; " n " C l; ; " a
m
:
Q- o O ^ o ^
(65)
o.
Simultaneous
Measurement
andJoint ProbabilityDistributions
in QuantumMechanics115
It is then a simple matter to show that (62) and (65) are sufficient for
the eaualitv
,t
r ll /s- Q^"i,1';@i2"- i;;^)(o,,,-o^)l (66)
(As/)2 -
(/o:./52 =- t -
mnr
mo
with iff;" given by (64a). From this (51) follows immediately, (64a) being the
value irf dffy"for which the minimum is attained.
An analogons relation obtained for AE, which has its minimum value
Ao0 when ifig", is given by (64b). From the fact that (64a)and (64b) ?r0 rot
allowed simultaneouslyii [1. B) + 0 it follows that the product A"e .lg
even obeysthe strong inequality
J.l
JyA-- Jt{
. lvq8
(67)
So our conclusion is that the measurementscheme (27) produces
measurement results which satisfy the Heisenberg uncertainty relations
whenever the scheme reproduces experimentally the quantum mechanical
averages.
We now investigatethe samequestion for the measurementscheme(44).
Here the phasecorrelation disturbanceresponsiblefor the ergodicity properties (46) may be describedby averagingthe scherne(27) over some ensemble
representingexternal fluctuations. When we expressthis averaging by (...)
and substitute (61) into (46a), we get
UI>>
KU I o^oxoto)(o-o,Xo6o
(tct:i"dXi;,.,>'u
) {*,,,><'lt^'n','I U' : 0,
I
m * mo
lnna
m'nt
ft'
Since {Ury'-,,} constitute a complete orthonormal set of functions in
Hilbert space,this has as a consequencethat
((.d1,,"dn9;,,,>t
:0,
mo{ mo
(68)
The only effect of this averaging procedure on the expressions(62),
(65), and (66) is the removal of nondiagonal terms. Therefore the general
conclusionsremain unchanged and the inequality (67) is a valid relation for
the measurementscheme(44) as well.
5.3. A Reflection on Feasibility
We conclude with a remark regarding the feasibility of the measurement
schemes(27) and (44) under the requirement (50). Although mathematically
n6
de Muynck, Janssen,and Santrnrn
the restrictionson dff;" displayedin (65) do not pose any problem. becausc
the macroscopicity of the measuringinstruments (representedby the parameter cv)guaranteesthe existenceof suffi.cientlymany degreesof freedom. lt l\
not so obvious how (65) should be realizedin a practical physicalsituution
How could macroscopicmeasuring instruments be constructedthat trb.-1
such subtle requirementsas (65)'?Of course this problem is not specificl'or
joint measurements.
Analogous questionscan be askedregardingthe theorr
of single measurements,where this problem obviously has been solred rn
practice for a number of observables.Now the problem seemsto be fur less
seriousfor the measurementscheme(M) than for (27), becausethe ar erlging
activity of the phase correlation disturbance in the case of (aa) has a' a
that the nondiagonalpart of (65) is satisfiedautomaticalll. i.e..
consequence
A"1ffi."ffi^);a,n:
o,
mo/' mn
becauseof (68). Thus it is seenthat the chance of successin constructing
joint measurementinstrumentsexperimentallylargelydependson the question
of whether (21) ot (44) is the correct scheme.
APPENDIX
A
In this appendix we prove a theorem concerning bounded Hermtttan
operators with a discrete spectrum. With a little effort it can be extended to
unbounded operators with a discretespectrum.
Theorem. Let A and -Bbe bounded, Hermitian operators with a di\crete
B: I b"E(N")" Let S be the largestsubspace
spectrum;A:La*E(M^),
(S
on which A and B commute C /f) and supposethat S reducesthe operators
A and B. Then
s-
UM*..Nn
m,n
Proof of the theorem. Define R:
@n,,nM^ A Nn, where tr[^ and
N, are the eigenspacesbelonging to the eigenvaluesa^ and b, of A and .8,
respectively.
(a) R is a subspace, since M* n ff" I M^, A N", (m + m' and or
n + n') and then it follows (Ref. 31, p. 42) that
R:
@M*^Nn:UM*^N,
n,n
tu,i
SimuhaneousMeasurementand Joint Probability Distributions in Quantum Mechanics 117
(b)
R reducesI and ,B since
V/eR
1:l.f-,,,
.f*,n e Mn, n N,
then
A.f :
V/e R
L Af^.n-:
I
a-J*.ne R
s i n c e f r o m J ) n , n €M - ^ I y ' , i t ; ; - ,
, h u r ' 7 ' ; , , " , ,M
, €, n , a n d c o n s e q r , : n t l y
Af,o.,: on,f,n,,,.So V/e R A.f e R, but then the subspaceR reducesthe
operator I since,4 is a bounded Hermitian operator (Ref.31, p. la9).
(c) The operators I and B commute on R while
Yf e M*n
Nn
lA, BIf :
[E(M*), E(N")]f :0
Since{M,, A Nn},n,,spansthe subspaceR, it follows that V/e R [A, B] f : 0.
SoRCS.
Subsequently we prove that R : S. To this end we suppose _rR
CS
and derive a contradiction. lf R C S, then E(S) - E(R) is the projection on
the subspaceS - R .,. S n R1 and it can be shown that the subspace,Sn Rr
reducesthe operators .4 and B. Furthermore, ,4 and _Bcommute on S n R L.
(d) It is not difficult now to prove (e.g., by a slight generalizationof
Ref. l, p. 170)that under the aboveconditionsthe spectralfamiliesof A and
-B commute on S n R-. r.e..
Vge SnRr
E(M*)E(N")C:
E(N"\E(M*)g
Choosea vector g e S n R-; then
1E(M^u), E(N,)
E(M,,,') E(N,,)g.- g(A,,,) E(Nf^")g + 0
(Al)
But then for all natural numbers /
{E(M",,) E(lf".)l t-t E(M^,) E(A',')g -- E(M,,,,) E(N,,,)g +. Q
since, according to (Al), E(M,,a)E(Nn)ge M,nu and E(M^)E(N,o)g 6 iy',o.
So
( E ( M - " ) E ( A ' " , u ) ) , g- 3
where 3>0
(e) lt is well known that(8'10,21)
E(M,,o A N,u) :
l;m {E(I[^,)
E(l/",)]t
(A2)
t
lrt
de Muynck, Jmssen, snd Ssnh8n
Then it follows from equality (A2) that
E(M-ooN")g*0
However, a contradiction is now obtained since on the one hand g e R - .S
impliesC I R and on the other hand (e) impliesC L R.Since this statement
is true for all g in S n R1, it follows that S A Ri : {0}. So ^S R.
APPENDIX B.
MARGENAU
THE COUNTEREXAMPLES
OF PARK AND
Park and Margenau (Ref.4, pp.239 ffconsider two experiments\\hrch
in their opinion representjoint measurementproceduresof incompatihle
observables:the "time-of-flight" method and an experimentwhich rie 'hall
call the "correlation" experiment. These procedures are characterized as
being of a "historical" type, in contrast with the procedurescorresponding
to our schemes(27') and (44), which are termed by them as "simple."
B.1. The "Time-of-Flight" Method
By means of an "electron gun" an ensembleof electronsis prepared
with state function
*(,,t): ({^1''' ,,pYt [ .^ol-# (","- +'-)]*,., o,i,r,i,
The state function ,!6, t - 0) is assumedto be of compact supprrrt.
i.e., l(x, t -.- 0) - 0 iff't f (-xo , xo).With this conditionPark and \largenr'u
show, using the usual quantum mechanicalrules for single measurenlen:.
that the probability for a measurementof momentum I to yieldp = | lt . l:)
at t - 0 equals the probability that a measurementof the obserrablenr.r''I
(in which .f is the position observable)yields mxlte(pt,p.:\ r\\ t - t.
Thus
Wulp e (p,., pr);r/(x, r : 0)l
: Wn,s,lfmxlt
e (p, , p); {(x, t); / *
rt]
(Bl)
The equality (B2) shows that the time-of-flight method prorides an
operational definition of a Z-measurementat t : 0 in terms of an "y'-measurementatl+cc.
Sincemomentum is conservedin free motion of the electron.the tblltrrri n g e q u a l i t yh o l d s :
Wrolp e (h , pr); /(x, 0)l
: lTelp e (h, pz); *(*, t)1,
t > 0
(B-1)
Simultaneous
Measurement
andJoint ProbabilityDistributionsin QuantumMechanics 119
From (82) and (83) it follows that
'o':':,::;i;);.1!!'
,);,* at
,";,',
ir,l'r,.,
(84)
This means that the probability for a measurement of 4 to yield
p e (h , p") for r > 0 equals the probability that a measurement of mff lt
yields mxlt e (pt , p) at t + co. From this Park and Margenau conclude
that the "time-of-flight" method provides an experimental procedu:e to
measuresimultaneouslythe observables? and .f. Consider the instant when
the electron is detected and a result for .9f emerges. For that instant, they
argue, one may concludewith full quantum mechanicaljustification that a
measurementof :'l would have yielded mxlt if the result of the measurement
of .f is x. To this conclusion Jauch(lo)has already objected by remarking
that the equality (83) "is not sufficient for assertingthat the actual value of
(7 at Ihe two times is equal," which objection also makes sensewhen we
read for "actual value" the "value obtained in a measurement."Under the
condition of compact support it only follows from (84) that the probability
distribution of ,7 may be calculated from that of .fl. However, (B4) does not
guarantee at all that a measurementresult x of ff implies a corresponding
value p : mxlt of lv for the same electron.
Of course,as long as this assertionis not verifiedor falsifiedby a measurement of g with a separate measuring instrument (which would be bound
to involve "simple" simultaneous measurement of I and ff), it could be
upheld in the way Park and Margenau do. When, however, an assertion is
fundamentally unamenableto experimental proof or disproof (a point taken
by Park and Margenau in this matter), it cannot be consideredto contribute
to scientificknowledgeand should be abandonned.
Sinceno joint probabilitydistributioncan be derived,Park and Margenau
postulatethat the following joint distribution holds for the "time-of-flight"
experiment:
L V ( xp
, ; *(x, t)):
i /(.r, t)t, 6(p -
mrlt)
(85)
This distribution implies again that Park and Margenau attribute a
valuep:mxlt
to that electronthat shows a value,x on measurementof .f
at the time /. Again, this conclusionis basedon the nonvalid inferencethat
from the equality of the distributionsof p and mxlt valuesto each electron
the tuple (x, p : mxlt) can be attributed.
Since the problem at issuein our opinion is the joint measurementof
two incompatible observableson one microsystemby two separatemeasuring
instruments, we conclude that the "time-of-flight" method does not provide
a method for simultaneous measurementof 5f and ?, although it provides
andSantman
de Muynck,Janssen,
120
an operational definition of Z-measurement in terms of ./-measurement.
Incidentally,we remark that an analogousstatementcan be made regarding
the experimental arrangement,proposed by Ballentine (Ref. 6, p. 36-5).
designedto simultaneouslymeasurey and p, . The joint probabilitl' dr'tribution corresponding to this arrangement should be proportional to
3(r - Q'tp) p,).
B.2. The Correlation Experiment
Consider two quantum systemsSr, and S, with observables.ry'r. 4,
and.il, associatedwith S, and Sr, respectively.Suppose[At, Br] -' 0 and
denote eigenfunctionsand eigenvaluesas follows:
Arof,i- all)"1."
A2"f' .,: o[""!!l
Let S, and S, be noninteracting, but in a correlated state
.l'-=L.u"f;)G)"l')
(86)
Now, traditionally the observables.cr'tand s/, are thought to be simult a n e o u s l ym e a s u r a b l es,i n c e
[ A r ,A r ) : [ ] , g l , I g A " l _ a
This also obtains for s/, and 9d, becauseof lAr,lr]_ .: 0 For measurement results of d and .{, the following holds:
w@ft), al'): ,l'\
=- c* '3.'
Tr Es,E"r6)3*1(z)
in which ,8, and E""rr)c.r",,r)
are the relevant projection operators. Thi\
implies that a measurementof .o/, on S, will yield rvith certaintl' al-' if a
measurementof s./, on the correspondingmicrosystemS, yieldedafr,.
Park and Margenau claim that a simultaneousmeasurementof .:/r an'J
./, is possibleas follows. Measure.&, on S, . This measurementdoe: not
involve interactionwith Sr, but provideswith certaintythe value that sould
have shown up in case of a measurement of s/, on Sr . Therefore. do not
m e a s u r e . d o n S , b u t i n s t e a dm e a s u r e9 r o n S r . A f t e r t h e m e a s u r e m e not f
s l r o n , S 2 a n d J B r o n S , o n e k n o w s w h a t r e s u l ta m e a s u r e m e not f . d r l o u l d
have yielded and what result the measurementof 4, in fact y'ielded.This.
accordingto Park and Margenau, is enough to considerthe procedureas a
s i m u l t a n e o um
s e a s u r e m e nstc h e m e .
The easiestway to show the inappropriateness
of this view is by'considering the next experiment.Preparean ensemblein an eigenstateof the operator
in QuantumMechanicsl2l
Measurement
Simultaneous
andJoint ProbabilityDistributions
l, . One may consider,for example,one of the two beams from a Stern-Gerlach device. After this, one can predict with certainty the result of a
measurementof the observable.c/, correspondingto the state preparation:
a second Stern-Gerlachdevice in the same tlirecti,rn as the first one will
produce only one beam. If, however, another observableis measured,say
;'r9,, one knows. according to the argument of Park and N{argenau, the
result of this .#r-measurementand one knows what a measurementof the
observablc'./, correspondingto the original state preparation would have
yieldeclif it hril beenperformed.So" if the operatorsl, and,8, do not,ommute, this rvould img,l',ra simultaneous measurement of incompatibie
observables.accordingro ihis view. This, however,is highly unusual.
That the situation rjescriLcdabove is relevantto the correlationexperiment of Park and Margcrratif,'llorvsii:.rp116" observationthat the measurement t-rf,/, oir "!, has the elTeclof a siittc:oreparaticlnfor S, . For, the correlation expresscdby Eq. (86) allows us to :;elec1[he subensembleof microsystemsS, correspondingto the subensemble
of microsystemsS, -vieldingon
measurementol ,r/, the result al2) (rvhich can be done for any k). This suhensembleis to be describedby the eigenfunctionaf') of l, . The measurement
of :1BronSr is, viewed in this light. to be consideredas a measurementperformed on these subensemblesand is rather to be characterizedas a conditiona! measurementprocedurethan as part of a joint measurementprocedure.
ACKNOWLEDGMENTS
The authors wish to thank Hans van den Berg, Roger Cooke, Jan
Hilgevoord, and Willem Roos for many interestingand fruitful discussions
and for critical reading of the manuscript. They also thank Boudewijn
Verhaar for some valuatrleremarks. Also, the inspiring criticism of one of
the refereesshould be acknowledeed.
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