78
This paper treats a class of codes made possible by
restrictions
principal.
on measurement related to the uncertainty
Two concrete examples and some general
results are given.
W
Conjugate Coding
Stephen Wiesner
Columbia university,
New York, N.Y.
Department of Physics
The uncertainty principle
capacity of certain
imposes restrictions
types of communication
paper will show that in compensation
channels.
for this
on the
This
"quantum noise",
quantum mechanics allows us novel forms of coding without
analogue in communication
channels adequately described by
classical physics.
Research supported in part by the National Science Foundation°
79
We w i l l
coding
and
first
give
then p r o c e e d
E x a m p l e One:
two m e s s a g e s
w h i c h m a y be
The
which
communication
polarized
the
light
light p i p e
all p o l a r i z a t i o n s
and
to a more
examples
abstract
of c o n j u g a t e
treatment.
channel
is a light pipe
is sent.
Since
the
or guide
information
down
will
be
in ths polarization, it is e s s e n t i a l
does
not
of light
depolarize
travel
with
the
the
light
same
and
that
velocity
attenuation.
The
binary
light
two m e s s a g e s
sequences.
at times
of the bursts
than
end
Before
chooses
a coin
one
of the
the
of the
in e i t h e r
whether
one.
of the
second
the
the
transmitter
then
sends
the
label
burst
form
will
of two
bursts
T 2, etco
The
it is unli]4ely
ith b u r s t
a bit
from
is chosen,
be d e t e c t e d
first
message
binary
a table
the
left-hand
of the
l, next
...),
of
amplitude
that
more
at the
by
flipping
numbers.
on w h e t h e r
is a zero
ith b u r s t
circular
second
manner
transmitter
sense
message
the
or a one.
is p o l a r i z e d
depending
is a zero
or a
sheet.
The
receiver
contains
fringent
crystal,
or
some
a quarter
other
wave plate
analyzer,
If
is p o l a r i z e d
depending
the
the
of r a n d o m
ith b u r s t
sequence
is chosen,
or
(i=1,2
in a r a n d o m
or h o r i z o n t a l l y
right
T I,
so that
two m e s s a g e s
ith d i g i t
See Fig.
the
light pipe.
emitting
vertically
digit
into
from e a c h
or s e l e c t i n g
either
ith
The
rendered
is a d j u s t e d
first message
the
are
that we w i l l
one p h o t o n
receiving
If
concrete
A means for t r a n s m i t t i n g
e i t h e r but not b o t h of
received.
conveyed by variat£ons
that
two
that
and b i r e separates
on
8O
POLARIZATION
OF ith BURST
- -
i th
- -
DIGIT
,
r
-
-
OF
FIRST SEQUENT, 7
VERTICAL
RAN
<•
_
_~....~...~.~_.~IpI
HORIZONTAL
i th DIGIT OF
•~ _ C O N D SEQUENCE
RIGHT
C LEFT
FIG, 1
81
orthogonally polarized
components
spatially separate beams.
Following
best available p h o t o m u l t i p l i e r
is to be received,
be received,
into
of the
If the first message
is arranged so as to send
to one phototube
to the other.
the separation
this is a pair
tubes°
the analyzer
v e r t i c a l l y p o l a r i z e d photons
p o l a r i z e d photons
of the light wave
and h o r i z o n t a l l y
If the second message is to
is made with respect
to right and
left-hand circular polarization.
Now if the linear p o l a r i z a t i o n
all chance of measuring
Thus,
of a p h o t o n
its circular p o l a r i z a t i o n
if the receiver is set to receive
Likewise,
when
is lost.
the first message,
nothing at all is learned about the contents
message.
is measured,
of the second
the receiver is set to receive
all information
concerning
the
second message,
it destroys
the
first message.
If the receiver is set up to sort the photons
with respect to some elliptical p o l a r i z a t i o n s
intermediate
between
linear and circqlar,
about each
message
is recovered
measurement
message,
than when
for the reception
Of course,
even when
a full k n o w l e d g e
recovered.
less information
the receiver makes
of one message
alone.
the receiver is set
of the
for the first
first sequence
In fact, half the digits of the
the best
is not
first sequence
never even influence
the transmitted signal and at the
corresponding
when
transmitted,
times,
the second message
is being
the receiver output has an equal p r o b a b i l i t y
of being a zero or a one.
coding
scheme,
photon
shot noise,
This noise i n t r o d u c e d by the
as well as the noise due to the channel,
and the p h o t o m u l t i p l i e r
noise,
may be
the
82
overcome
used
if an error
in forming
messages.
would
correcting
the b i n a r y
code of the usual
sequences
Care must be taken,
allow both messages
reception
of one s e q u e n c e
There
is no w a y
for too much r e d u n d a n c y
to be r e c o v e r e d by the a l t e r n a t e
and then
complete
contents
messages
so long as it is c o n f i n e d
on one burst
exist
all
of p h o t o n s
the t r a n s m i t t e d
transmission
vector
information.
light
For
follows
consider
length.
of
the
The trans-
the set
pairs
exists
corresponding
an H e r m e t i a n
to a d i f f e r e n t
c o u l d be
of the type p r e v i o u s l y
that was
Y must be o r t h o g o n a l
of the v e c t o r s
operator
space
If one of the
from Y to Y'
the m e s s a g e
of finite m e s s a g e s
operators
measurements
[Y]
signal may be
transmissions.
by a r e c e i v e r
if set to r e c e i v e
that
Hermetian
To see this,
~ in a large H i l b e r t
The change
to be possible,
possible
there
a state
unambiguously
this
there
that a l l o w r e c o v e r y
of finite
finite
4' is p r o d u c e d .
described,
coded
measurements
In principle,
and the entire
by a single v e c t o r
is changed,
detected
to m a k i n g
measurements
s p a n n e d by all p o s s i b l e
messages
the
than one of the c o n j u g a t e l y
of two m e s s a g e s
of p o l a r i z e d
described
can r e c o v e r
produce a signal c o n s i s t i n g of a finite n u m b e r of
m i t t e r will
bursts
the other.
at a time.
very c o m p l i c a t e d
is
from the o r i g i n a l
that the r e c e i v e r
of more
sort
changed.
to Y'.
corresponding
is o r t h o - n o r m a l
It
to all
and t h e r e f o r e
or a set of c o m m u t i n g
corresponding
to a m e a s u r e m e n t
or
that can d i s t i n g u i s h
all the p o s s i b l e
signals.
83
There is an easy extension
to the case of three messages,
no two of which may be recovered.
third binary sequence using
One simply transmits
light in the two p o l a r i z a t i o n
states at 45 ° to vertical and horizontal.
than three messages
is not straight
The above system for sending
Extension
two m u t u a l l y exclusive
could be built at the p r e s e n t
possible
in p r i n c i p l e
time.
to beat the system and recover both
c o m p l e t e l y beyond the reach of p r e s e n t - d a y
next example
that are
technology.
The
in p r a c t i c e but not in principle~
is in the opposite
in principle,
present
Though it is
to do so w o u l d require measurements
system therefore works
to more
forward.
messages
messages,
a
category;
The
it is foolproof
but it p r o b a b l y could not be built at the
time.
Example Two:
Money that it is p h y s i c a l l y
impossible to counterfeit.
A piece
of quantum m o n e y will contain a number of
isolated two-state p h y s i c a l
isolated nuclei
of spin 1/2.
let a and b represent
let ~ = i//2(a+b)
systems
such as,
for example,
For each two-state
a pair of ortho-normal base
and ~ = i//2(a-b)
the useful
the orthogonal
and
isolated
from
so that if one of them is i n i t i a l l y
in the state a or ~, there is little chance
made during
states
represent a n o t h e r pair.
The two state systems must be well enough
the rest of the universe
system,
lifetime
that a m e a s u r e m e n t
of the money will
states b or ~, respectively.
find it in
There is no
device
operating
of a t w o - s t a t e
second;
will
system
however,
surely
at p r e s e n t
change
isolated
t h e y create
is p r e s e r v e d
the c o n t i n u i n g
the
"phase
for longer
advance
to be d e f i n i t e
systems,
S i, i=l,
two r a n d o m b i n a r y
than about
of c r y o g e n i c
that the m o n e y
2,
...
sequences
20.
N i = 0 or 1.
the t w o - s t a t e
four states
scheme
shown
technique
systems
M i = 0 oK i,
are p l a c e d
a, b, ~ or ~ in a c c o r d a n c e
in Fig.
contains
of t w e n t y digits
call M i and N i, i=i~2,...20,
of the
a
At the mint
e a c h w h i c h we will
Then
coherence"
this.
Let us suppose
twenty
84
in w h i c h
with
in one
the
2.
a
0
b
I
FIG. 2
The m o n e y
printed
is also given
a serial
number which
on it in the u s u a l w a y and the two b i n a r y
is
sequences
s5
describing
its
and p e r h a p s
When
made
the m o n e y
of q u a n t u m
he
not k n o w
make
on S l..
money.
M i,
Likewise
destroys
the
goes
S i and p r o d u c e s
there
make
of e a c h
Then
whole
that
counterfeit
measurements
makes
i,
found wrong
and
Suppose
states
on
found
is a 50% c h a n c e
in
this
at the
is
event,
mint will
a 1/4
is o n l y
show
chance
the p r o b a b i l i t y
inspection
~ from
measurement
the
there
there
to
of d i s t i n g u i s h i n g
some
and
since
distinguishes
a from b.
a measurement
coin passing
S . that
the n e w S i in
Thus,
duplicate
distinguishes
measurement
state.
state.
N i because,
chance
that
for e a c h
the w r o n g
being
all
anyway,
money with
in the w r o n g
digit
what
on a p a r t i c u l a r
ahead
is a 50% c h a n c e
S i to be
not k n o w
is
initial
who would
recover
of d i s t i n g u i s h i n g
measurements.
he w i l l
cannot
in its
orthogonal
of s o m e o n e
destroy
the m i n t
a check
is still
to the
a measurement
chance
a counterfeiter
that
does
at
banks.
system
He
necessarily
f r o m ~.
by his
he
on r e c o r d
to the mint,
switched
A measurement
a from b must
the
isolated
the p r o b l e m
a piece
does
of b r a n c h
it has
consider
are k e p t
is r e t u r n e d
if e a c h
or w h e t h e r
Now
state
at a n u m b e r
to see
state
initial
of
the
(3/4) 20 <
0. 00317.
Could
learning
made
there
the
can be
But
there
can be m a d e
an u n l i m i t e d
supply
determined.
this
is
some
sequence
(so that
copies
be
way
N i.
are
of d u p l i c a t i n g
No,
two p i e c e s
by making
copies
of s y s t e m s
Thus,
impossible.
because
the
the
if one
money
copy
of the money)
of c o p i e s .
in the
sequence
same
N i could
can be
then
Now
state,
be
without
many
given
that
state
recovered.
86
Conjugate
If the m o m e n t u m
known
about
wise,
of a p a r t i c l e
its position;
to be found
if the p o s i t i o n
conjugate
possessing
is known,
of c o n j u g a t i o n
and this
it is e q u a l l y
a fixed volume
holds
suggests
from v a r i a b l e s
then n o t h i n g
then nothing
The same r e l a t i o n
variables
is known,
in other words,
in all regions
its momentum.
Bases
Like-
about
all pairs
an e x t e n s i o n
to basis
likely
V.
is k n o w n
between
is
of
of the idea
sets.
Let
[ai],
be
i=l,2,...,N
two o r t h o - n o r m a l
We call
such a p a i r
for all i and j.
described
and vice versa,
Physically,
A collection
of bases
present
systems
case w h e r e
the above
This
not
*
found
space.
it_must
is in a state
have
an equal
found in any of the states
to be
b
hi, i=l, ...,N
it must h a v e
l
an
in any a •
1
of bases will
be called
conjugate
is conjugate.
code
is any c o m m u n i c a t i o n
used
as signals
to e l e m e n t s
describing
entire
Hilbert
if each
We can n o w
a definition.
physical
space
then
in the c o l l e c t i o n
A conjugate
ponding
if a system
if it is in a state
equal p r o b a b i l i t y
i=l, ....N
if and only if I (ai,bj)l 2 = iN
conjugate
of being
[hi],
for an N d i m e n s i o n a l
by a i, i=l, ...,N,
probability
pair
bases
and
the individual
the sequence
definition
transmissions
last
of several
condition
are p l a c e d
basis
systems.
Note
of signals
has more
to be elements
fulfilled
in w h i c h
in states
conjugate
does not require
was
scheme
corres-
of the H i l b e r t
that
than
of c o n j u g a t e
in the
one e l e m e n t
the vectors
in the second
the
describing
base
sets.
example
but
in the first.
(ai,/b i) is the
inner product <ailbi > in the D i r a c
notation.
87
In a d d i t i o n
of c o n j u g a t e
to p a i r s
bases.
of c o n j u g a t e
For example,
bases,
there
are triplets
in a t w o - d i m e n s i o n a l
system
we have
[~, b]
lal2:Ibl2=l
(a,b)=0
[i/72 (a+b), i/72 (a-b) ]
{i//2(a+ib) , 1//(a-ib)]
Three
such bases were
three m e s s a g e s
Are
theorem
mutually
sets b i g g e r
conjugate
exists
Suppose-the
basis
than t r i p l e t s ?
In an H i l b e r t
to the m u l t i p l i c i t y
space
of d i m e n s i o n
of N m u t u a l l y c o n j u g a t e basis sets.
be
t h e o r e m to/true for N ~ M / / o
Let
space H of dem°
i=l
of
2 (N-I) ;/2,
sets
As[ a is]
conjugate
ortho-normal
Proof:
basis
on an
2(M-l) :/2 ~ D
.o. D
and
2
, a
for all ~ # 6
)i
o
We can then c o n s t r u c t
the
space H
For
the
basis
~
H
~
~.o
first M basis,
sets A s.
the v e c t o r s
*@
The f o l l o w i n g
sets.
C~=i...M be a set of m u t u a l l y
Hilbert
for sending
can be received.
that there is no limit
Theorem:
there
used in the scheme
no two of w h i c h
there
shows
o
aI
M+i m u t u a l l y
O
we
H = H M.
s
x~
a3
•
s
o
•
i
is the tensor product.
of all linear functions
~
set
az s
bases
on
*
take a n a t u r a l
Call A ~ the basis
conjugate
extension
of the
of H M c o n s i s t i n g
i,
j,
•
~
•
H x ~ H' is d e f i n e d
from H into H'.
~
~
~
o
of
D
o
as the space
88
Note
that is ~ ~ ~,
2
c~, am ~
m~ ) I2
=
X
M
(a z~, ap ~) I
---X
so these basis
sets
1
= 7~
here,
n+JG Mod(D)
Since
we take the vectors
D
27Ti ~
E e
D
K=i
q = 1 ... D and
permutations
(})
[A~] are m u t u a l l y conjugate.
For the last basis,
V(q'[P~])
=
1
x aK
[P~],
@
2
ap2(K)... ~
M
apM(K )
~ = 2, 3 ... M is a set of cyclic
on the integers
1 ... D.
(i.e., P~(n)
=
for some integer J .) Call this last basis V.
there are D cyclic p e r m u t a t i o n s
there are D M-I sets
on D intergers,
[P~] and DxD M-1 = D M v e c t o r s
V(q,[P~])
in
V; as there should be.
The p r o o f that V is o r t h o - n o r m a l
So,
actually,
other basis sets,
matter.
of A s.
is the p r o o f
that V is c o n j u g a t e
to the
but I give it since it is the h e a r t
Fix ~ and let W m a i ~
...~
aZ
of the
be a typical v e c t o r
Then
D
27ri q K
D
I (W,V(q, [PSI) )12 = ~I Z e
x
K=i
(ai(Z~
The inner p r o d u c t will be zero unless
term of W.
(Let P
1
lID1 (ai ®
~
... a~ , ak ( D " "
the ~th v e c t o r
product.
'
Then
M
~
1
M
... az 'aK Q ' ' ' a P M(K) ) 1
ap~(K ) equals
be the identity.)
one value of K, call it k.
where
is obvious.
This h a p p e n s
for just
I (W,V(q, [P~]))I 2 =
apM(k))l
2
is the same on both sides
As for the rest,
the ~th
of the inner
I (ai ~, a~P~k)) 12 = i/D
