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Volume 151, number 3,4
PHYSICS LETTERS A
3 December 1990
Incompatible results of quantum measurements
Asher Peres
Department of Physics, Technion-lsrael lnstitute of Technology, 32 000 Haifa, Israel
Received 28 August 1990; accepted for publication 4 October 1990
Communicated by J.P. Vigier
Quantum theory is incompatible with the following propositions. ( 1) The result of the measurement of an operator A depends
solely on A and on the system being measured. (2) If operators A and B commute, the result of a measurement of their product
AB is the product of the results of separate measurements of A and orB.
Various q u a n t u m " p a r a d o x e s " [ 1-5 ] are based on
the a s s u m p t i o n that the result o f the m e a s u r e m e n t o f
an o p e r a t o r A d e p e n d s only on A and on the state o f
the q u a n t u m system being measured (here, the word
" s t a t e " includes not only the wave function ~u, but
also any h i d d e n variables that theorists m a y invent).
In particular, that result does not d e p e n d on the
choice o f other m e a s u r e m e n t s that m a y also be performed on distant physical systems. The purpose o f
this Letter is to give an elementary p r o o f that these
assumptions are i n c o m p a t i b l e with q u a n t u m theory.
Consider two spin i particles in a singlet state. The
result o f a m e a s u r e m e n t o f tz~x (which may be only
_+ 1 ) will be called xj, and similar notations used for
the other components. We m a y measure a product
a~xa2:, either by measuring thx and 0"2xseparately, with
result x~x2, or in a single nonlocal procedure. The
result o f the latter must be - 1 ,
because
( t r l x t r 2 x ) = - I for a singlet state. It follows that
x j x 2 = - 1. Likewise YlY2= - 1 and z~z2= - 1.
Consider now a m e a s u r e m e n t o f a~xa2y. We can
measure tr~x and a2y separately, with results x~ a n d
Y2. We assume that these are the same x~ and Y2 as
m e n t i o n e d above - that is, the result o f measuring
a~x does not d e p e n d on whether the other measurem e n t is (was, will b e ) a m e a s u r e m e n t o f a2:, or one
o f a2y. Therefore, if we perform a direct measurement o f t71xa2y, the result is xly2.
Note, however, that
[a,xa~y, a,ya2x] = 0 ,
so that we can measure trtya2x together with O'lxO'2y ,
without mutual disturbance. ( O f course, we cannot
measure a~ytr2x together with trtx and a2y separately. )
The result o f measuring a~ya2~ is y~x2, by the same
argument as above.
Now, we also have, for the product o f these two
c o m m u t i n g operators,
O'lxa2yO'lya2x = Olx O'ly 0"2y O'2x ~---alzt72z,
(2)
whose expectation value is (a~ za2~) = -- 1. It follows
that
Xl Y2Yl x2 = ( alxa2~aly a2x ) = - 1 ,
(3)
in contradiction with x~x2 =Y~Y2 = - 1.
This simple exercise shows that what we call "the
result o f a m e a s u r e m e n t o f A" cannot d e p e n d only
on A and on the state o f the system (unless the wave
function is an eigenstate o f A ) . It also depends, in a
way not yet understood, on the choice o f other quant u m measurements that m a y possibly be performed.
This property could also be inferred from the references listed below. However, the p r o o f given here
is much simpler than that o f Kochen and Specker
[3 ] (who found a similar incompatibility among 117
measurements o f a particle o f spin 1 ) a n d the more
recent proofs in refs. [4] and [5], which require four
and three particles, respectively, instead o f just two,
as here.
( 1)
0375-9601/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
107
Volume 151, number 3,4
PHYSICS LETTERS A
Note added. It was shown by M e r m i n [ 6 ] that the
use o f a singlet state is not essential to the above argument: Consider the nine operators
alx
G2x
~lxa2x
0"2.v
O]y
0"lyO'2y
£YlxO'2y ~lya2x
This work was s u p p o r t e d by the G e r a r d Swope
Fund, and by the F u n d for Encouragement of
Research.
References
alz~2z .
The three operators in each row and each column
commute, and their product is 1, except those o f the
last column, whose product is - 1 . There is obviously no way o f assigning numerical values + 1 to
these nine operators, with this multiplicative property. I a m grateful to N.D. M e r m i n for a stimulating
correspondence.
108
3 December 1990
[ 1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 (1935)
777.
[ 2 ] J.S. Bell, Physics 1 ( 1964 ) 195.
[ 3 ] S. Kochen and E.P. Specker, J. Math. Mech. 17 ( 1967 ) 59.
[4] D.M. Greenberger, M. Home and A. Zeilinger, in: Bell's
theorem, quantum theory, and conceptions of the universe,
ed. M. Kafatos (Kluwer, Dordrecht, 1989) p. 69.
[5 ] N.D. Mermin, Phys. Today 43 (June 1990) 9.
[6] N.D. Mermin, preprint.