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Ask a question or post a comment about the paper
Physics
Vol.
1,
No.
3,
pp.
195-290,
1964
Physics
Publishing
Co.
Printed
in
the
United
States
ON
THE
EINSTEIN
PODOLSKY
ROSEN
PARADOX*
].
S.
BELLt
Department
of
Physics,
University
of
Wisconsin, Madison, Wisconsin
(Received
4 November 1964)
I.
Introduction
THE
paradox of
Einstein,
Podolsky
and
Rosen
[1]
was
advanced
as
an argument
that
quantum
mechanics
could
not
be
a
complete
theory
but
should
be
supplemented
by
additional
variables.
These
additional
vari-
ables
were to
restore
to
the
theory
causality
and
locality
[2]. In
this
note
that
idea
will
be
formulated
mathematically
and shown to
be
incompatible
with
the
statistical
predictions
of
quantum
mechanics.
It
is
the
requirement
of
locality,
or
more
precisely
that
the
result
of
a
measurement
on one
system
be
unaffected
by
operations
on a
distant
system
with which
it
has
interacted
in
the
past,
that
creates
the
essential
dif-
ficulty.
There
have
been
attempts
[3]
to
show
that
even
without
such
a
separability
or
locality
require-
ment
no
"hidden
variable"
interpretation
of
quantum
mechanics
is
possible.
These
attempts
have
been
examined
elsewhere
[ 4] and found wanting. Moreover, a hidden
variable
interpretation
of
elementary
quan-
tum theory
[S]
has
been
explicitly
constructed.
That
particular
interpretation
has
indeed
a
grossly
non-
local
structure.
This
is
characteristic,
according
to
the
result
to
be
proved
here,
of
any
such
theory which
reproduces
exactly
the
quantum
mechanical
predictions.
II.
Formulation
With
the
example
advocated
by Bohm and Aharonov [6],
the
EPR
argument
is
the
following.
Consider
a
pair
of
spin
one-half
particles
formed somehow
in
the
singlet
spin
state
and moving
freely
in
opposite
directions.
Measurements
can
be
made,
say
by
Stern-Gerlach
magnets,
on
selected
components
of
the
Spins
d l
and
a
2
,
If
measurement
Of
the
component d I '
a,
where a
is
some
unit
vector,
yields
the
value
+ 1
then,
according
to quantum
mechanics,
measurement
of
a
2
·a
must
yield
the
value
-1
and
vice
versa.
Now we make
the
hypothesis
[2],
and
it
seems
one
at
least
worth
considering,
that
if
the
two
measure-
ments
are
made
at
places
remote from
one
another
the
orientation
of
one
magnet
does
not
influence
the
result
obtained
with
the
other.
Since
we
can
predict
in
advance
the
result
of
measuring
any
chosen
compo-
nent
of
a
2
,
by
previously
measuring
the
same
component
of
d
1
,
it
follows
that
the
result
of
any
such
measurement
must
actually
be
predetermined.
Since
the
initial
quantum
mechanical
wave
function
does
not
determine
the
result
of
an
individual
measurement,
this
predetermination
implies
the
possibility
of
a more
complete
specification
of
the
state.
Let
this
more
complete
specification
be
effected
by
means
of
parameters
A.
It
is
a
matter
of
indiffer-
ence
in
the
following
whether
A.
denotes
a
single
variable
or a
set,
or
even
a
set
of
functions,
and whether
the
variables
are
discrete
or
continuous.
However, we write
as
if
A were a
single
continuous
parameter.
The
result
A
of
measuring
a
1
·a
is
then
determined by a and
A.,
and
the
result
B
of
measuring
7,
2
• b in
the
same
instance
is
determined
by b and
A.,
and
*Work
supported
in
part
by
the
U.S.
Atomic
Energy
Commission
tan
leave
of
absence
from
SLAC
and
CERN
195
196
].
S.
BELL
Vol.
1,
No.
3
A
(l:i,
,\)
= ±
1,
B
(b,
,\)
= ± 1.
(1)
The
vital
assumption
[2]
is
that
the
result
B for
particle
2
does
not
depend
on
the
setting
;;,
of
the
magnet
for
particle
1, nor A on b.
If
p (,\)
is
the
probability
distribution
of
,\
then
the
expectation
value
of
the
product
of
the
two com-
~ ~
~
7
ponents
o
1
•
a
and
o
2
• D
is
P
(t;,
b)
= .fn p
(,\)A
(~,
,\) B
(b
,
,\)
(2)
This
should
equal
the
quantum
mechanical
expectation
value,
which for
the
singlet
state
is
(3)
But
it
will
be
shown
that
this
is
not
possible.
Some might
prefer
a formulation in which
the
hidden
variables
fall
into
two
sets,
with
A
dependent
on
one
and
B on
the
other;
this
possibility
is
contained
in
the
above,
since
,\
stands
for any number
of
vari-
ables
and
the
dependences
thereon
of
A
and
B
are
unrestricted.
In a
complete
physical
theory
of
the
type
envisaged
by
Einstein,
the
hidden
variables
would
have
dynamical
significance
and
laws
of
motion;
our
,\
can
then
be
thought
of
as
initial
values
of
these
variables
at
some
suitable
instant.
111.
111
us trot
ion
The
proof of
the
main
result
is
quite
simple.
Before
giving
it,
however,
a number
of
illustrations
may
serve
to
put
it
in
perspective
.
Firstly,
there
is
no
difficulty
in
giving
a
hidden
variable
account
of
spin
measurements
on a
single
particle.
Suppose
we
have
a
spin
half
particle
in a pure
spin
state
with
polarization
denoted
by a
unit
->
_.
vector
p.
Let
the
hidden
variable
be
(for
example)
a
unit
vector
,\ with uniform
probability
distribution
over
the
hemisphere
A · p > 0.
Specify
that
the
result
of
measurement
of a
component
; · ;;
is
->
-> I
sign
,\ · a ,
(4)
where
-.;i
is
a
unit
vector
depending
on
~
and
p
in
a way
to
be
specified,
and
the
sign
function
is
+ 1 or
-1
according
to
the
sign
of
its
argument.
Actually
this
leaves
the
result
undetermined
when ,\ ·
a'_.=
0,
but
as
the
probability
of
this
is
zero
we
will
not
make
special
prescriptions
for
it.
Averaging
over
,\
the
expectation
value
is
< ; ·
~
> = 1 - 2
e'
Irr
,
(5)
where
()
1
is
the
angle
between
t;'
and
p.
Suppose
then
that
"ii'
is
obtained
from
; by
rotation
towards
P
until
2
()'
1 -
TT
cos
()
-> ->
where
()
is
the
angle
between
a
and
p.
Then
we
have
the
desired
result
< ; .
;;
> =
cos
()
(6)
(7)
So in
this
simple
case
there
is
no
difficulty
in
the
view
that
the
result
of
every
measurement
is
determined
by
the
value
of
an
extra
variable,
and
that
the
statistical
features
of
quantum
mechanics
arise
because
the
•
value
of
this
variable
is
unknown
in
individual
instances.
Vol.
1,
No.
3 ON
THE
EINSTEIN
PODOLSKY
ROSEN
PARADOX
197
Secondly,
there
is
no
difficulty
in reproducing, in
the
form
(2),
the
only
features
of
(3)
comm
only
used
in
verbal
discussions
of
this
problem:
(
.... ....
(
....
....
P a, a) = - P a, -
a)
= - 1 l
P
<a,
b)
= o
if
;; .
1i
= o
~
(8)
For
example
,
let
A now
be
unit
vector
A,
with uniform
probability
distribution
over
all
directions,
and
take
This
gives
A(a,
A)
=
sign
a.
A t
B (a, b) = -
sign
b · A \
P
<a
,
b)
= - 1 +
3.
e ,
Tr
(9)
(10)
where e
is
the
angle
between
a ·
and
b,
and (10)
has
the
properties
(8).
For
comparison,
consider
the
re-
sult
of
a modified theory [6]
in
which
the
pure
singlet
state
is
replaced
in
the
course
of
time
by
an
iso-
tropic
mixture
of
product
states;
this
gives
the
correlation
function
(11)
It
is
probably
less
easy,
experimentally,
to
distinguish
(10) from (3), than (11)
from
(3).
Unlike (3),
the
function (10)
is
not
stationary
at
the
minimum
value
-
l(at
e = 0).
It
will
be
seen
that
this
is
characteristic
of
functions
of
type
(2).
Thirdly,
and
finally,
there
is
no
difficulty
in
reproducing
the
quantum
mechanical
correlation
(3)
if
the
results
A and B in (2)
are
allowed
to
depend
on
b and a
respectively
as
well
as
on a
and
b.
For
ex-
ample,
replace
a in
(9)
by
a',
obtained
from
a
by
rotation
towards
b
until
2 I
1 - - e =
cos
e,
Tr
where
e'
is
the
angle
between
a'
and
b.
However, for given
values
of
the
hidden
variables,
the
results
of
measurements
with one
magnet
now
depend
on
the
setting
of
the
distant
magnet, which
is
just
what
we
would
wish
to
avoid.
IV.
Contradiction
The
main
result
will
now
be
proved.
Because
p
is
a
normalized
probability
distribution,
/d>..p(A.) =
l,
(12)
and
because
of
the
properties
(1), p
in
(2)
cannot
be
less
than
- 1.
It
can
reach
- 1
at
a = b only
if
A
(a,
>..)
= - B
(a,
A)
except
at
a
set
of
points
A
of
zero
probability.
Assuming
this,
(2)
can
be
rewritten
P(a,
b)
= - fa>..p(A.) A(a,
A)
A(b,
A).
(13)
(14)
198
].
S.
BELL
It
follows
that
c
is
another unit
vector
using
(1), whence
P(a,
b)
-P(;,
c) = - fi1i.p(A)
[A(a,
A)
A(b,
A)
-A(a,
A)
A(c,
,\))
= fi1i. p
(A)
A
(a,
,\)
A
(b,
,\)
[A
(b,
,\)
A
(c,
,\)
-1)
1 P(a,
"h)
-P(a,
c) 1 s f<np(,\)
[1-
A(b,
,\)
A(c,
1i.)J
The
second
term on
the
right
is
P
(b,
c),
whence
1 + p
<"h,
c)
2 1 P
<a,
6)
- P
<a,
c)
1
Vol.
1,
No.
3
(15)
Unless
p
is
constant,
the right hand
side
is
in general
of
order I
b-
c I for
small
I
b-
c I .
Thus
p
(b,
c)
cannot
be
stationary
at
the
minimum value
(-1
at
b =
c)
and
cannot
equal the quantum
mechanical
value (3).
Nor
can
the quantum
mechanical
correlation (3) be arbitrarily
closely
approximated by
the
form
(2).
The
formal proof
of
this
may be
set
out
as
follows.
We
would not worry about failure
of
the
approximation
at
isolated
points,
so
let
us
consider
instead
of (2) and (3) the functions
P(a,
6)
and
_;;;.
Ii
d
. d d . f
p (
....
,
71)
d
->1
71
->1
d
71
. •
where
the
bar
enotes
1n
epen
ent
averaging
o
a,
o an
-a
· o over vectors a an o within
sp~c-
ified
small
angles
of a
and
b.
Suppose
that
for
all
;; and b the difference
is
bounded by
€:
-
1 p
<a,
b)
+
-a
. 6 1 s f
Then
it
will be shown
that
€
cannot
be made arbitrarily small.
Suppose
that
for
all
a and b
Then
from
(16)
IPC.i,
t,)
+;;;·"his
(+a
From (2)
p
<a,
b)
=
j:i
1i.
p
(,\)
,4
<a,
,\)
'B
(b,
,\)
where
I A
<a,
;..)
I s 1
and
I
'B
<6,
;..)
I s 1
From
(18) and (19), with ;; =
b,
From (19)
I
d,\p(A)
[A(b,
A)
B(b,
A)
+
1)
s ( + 8
p
<a,
6)
.:.
P
<a,
c)
=fa
1i.p
<1i.)
[A
<a,
;..)
'B
<"h.
1i.)
-
.4
<Ii.
1i.>
'B
<c,
1i.)J
=
f:.>.p<1i.>
.4<8,
>.)
'B<Ti.
1i.)
(1+.4<".b.1i.)
s<c,
A)J
-I'>.p(A)
A<a,
;..)
'B<c,
A)
(1
+
.4ct,,
>.)'Bet,,>.)]
(17)
(18)
(19)
(20)
(21)
Vol.
1,
No.
3
Using
(20) then
Then
using
(19) and 21)
Finally,
using
(18),
or
ON
THE
EINSTEIN
PODOLSKY
ROSEN
PARADOX
1
?c:;,
b)
-?c:;,
;;)1
.'.S
;:11
...
ru,)
[1 +
A"ci,
A)
8c;;,
A)]
+
/aAp(A)
[1 +
A(b,
A)
B(b,
A)]
I
:;
·
;;
-
:;
·
"b
I - 2
(E
+
o)
< 1 -
"b
·
;;
+ 2
(E
+ o)
~
~
~
7 7
~
4
(E
+
0)
2:
I a · c - a · b I + b • c - 1
Take
for example a ·
(;
=
0,
a · b = b ·
(;
= l/
y'2
Then
4
CE
+ o)
;::,
v2 - 1
Therefore,
for
small
finite
o,
€
cannot
be
arbitrarily
small.
199
(22)
Thus,
the
quantum
mechanical
expectation
value
cannot
be
represented,
either
accurately
or arbitrar-
ily
closely,
in
the
form
(2).
V.
Generalization
The
example
considered
above
has
the
advantage
that
it
requires
little
imagination
to
envisage
the
measurements
involved
actually
being
made. In a more formal way,
assuming
[7]
that
any Hermitian oper-
ator
with a
complete
set
of
eigenstates
is
an
"observable",
the
result
is
easily
extended
to
other
systems.
If
the
two
systems
have
state
spaces
of
dimensionality
greater
than 2 we
can
always
consider
two dimen-
sional
subspaces
and
define,
in
their
direct
product,
operators
d
1
and d
2
formally
analogous
to
those
used
above
and which
are
zero
for
states
outside
the
product
subspace.
Then for
at
least
one
quantum
mechanical
state,
the
"singlet"
state
in
the
combined
subspaces,
the
statistical
predictions
of quantum
mechanics
are
incompatible
with
separable
predetermination.
VI.
Conclusion
In a theory
in
which
parameters
are
added
to quantum
mechanics
to
determine
the
results
of
individual
measurements,
without
changing
the
statistical
predictions,
there
must
be
a mechanism whereby
the
set-
ting
of
one
measuring
device
can
influence
the
reading
of
another
instrument, however remote. Moreover,
the
signal
involved
must
pr~pagate
instantaneously,
so
that
such
a theory
could
not
be
Lorentz
invariant.
Of
course,
the
situation
is
different
if
the
quantum
mechanical
predictions
are
of
limited
validity.
Conceivably
they might apply only to
experiments
in
which
the
settings
of
the
instruments
are
made
suffi-
ciently
in
advance
to
allow
them
to
reach
some mutual rapport by
exchange
of
signals
with
velocity
less
than or
equal
to
that
of
light.
In
that
connection,
experiments
of
the
type
proposed
by
Bohm
and
Aharonov
[6],
in
which
the
settings
are
changed
during
the
flight
of
the
particles,
are
crucial.
I
am
indebted
to Drs.
M.
Bander
and].
K. Perring for
very
useful
discussions
of
this
problem. The
first draft
of
the
paper
was written during a
stay
at
Brandeis University; I
am
indebted
to
colleagues
there
and
at
the University
of
Wisconsin
for
their
interest
and
hospitality.

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