VOLUME

70,

NUMBER 13

PHYSICAL

REVIEW

LETTERS

29 MARCH

1993

wave

plates

will

perform

these

unitary

operations.

)

Thus

an

accurate

teleportation

can

be

achieved

in all

cases

by

having

Alice tell

Bob

the classical outcome of her

mea-

surement,

after which Bob

applies

the

required

rotation

to transform the state

of

his

particle

into a

replica

of

IP).

Alice,

on the other

hand,

is left with

particles

1 and 2 in

one of the

states

4~2

)

or

IC~~

),

without

any

trace

of

(+)

'

(+)

the

original state

I

P)

.

Unlike the

quantum

correlation of

Bob's

EPR

particle

3 to

Alice's

particle

2,

the result

of

Alice's

measurement

is

purely

classical

information,

which can be

transmit-

ted, copied,

and stored at

will

in

any

suitable

physical

medium. In

particular,

this information need not

be

de-

stroyed

or canceled

to

bring

the

teleportation

process

to

a

successful conclusion:

The

teleportation

of

lg)

from

Alice

to

Bob has the side effect of

producing

two bits of

random classical information, uncorrelated to

lg),

which

are left behind

at

the

end of the

process.

Since

teleportation

is

a

linear

operation

applied

to

the

quantum

state

IP),

it

will

work

not

only

with

pure

states,

but also

with mixed or

entangled

states. For

example,

let

Alice's

original particle

1

be

itself

part

of an

EPR

singlet

with another

particle,

labeled

0,

which

may

be

far

away

from

both

Alice

and Bob.

Then,

after

teleportation,

particles

0

and 3

would

be

left in a

singlet state,

even

though

they

had

originally

belonged

to

separate

EPR

pairs.

All

of

what we have

said above can

be generalized

to

systems

having

N

&

2

orthogonal

states. In

place

of an EPR

spin pair

in the

singlet state,

Alice would

use a

pair

of

N-state

particles

in

a

completely

entangled

state. For

definiteness let us

write this

entangled

state

as

P

.

I j) Ij

)

/~N,

where

j

=

0,

1,

. . .

,

N

—

1

labels the

N

elements

of an orthonormal

basis for each of

the

N-state

systems.

As

before,

Alice

performs

a

joint

measurement

on

particles

1 and

2.

One

such measurement

that has

the

desired effect is

the one whose

eigenstates

are

lg„),

defined

by

lg„~)

=

)

e

"'~"

j)

S

I(j

+

m)

mod

N)

/~¹

(lu2)

I»)

+

I») lqs))

(9)

where

(Iu), lv))

and

(Ip),

Iq))

are

any

two

pairs

of

or-

thonormal

states.

These are

maximally

entangled

states

[ll],

having

maximally

random

marginal

statistics

for

measurements

on

either

particle separately.

States

which

are

less

entangled

reduce

the

fidelity

of

teleportation,

and/or

the

range

of

states

lg)

that can

be accurately

tele-

ported.

The

states

in

Eq.

(9)

are also

precisely

those

ob-

tainable

from

the EPR

singlet

by

a

local

one-particle

uni-

tary

operation

[12].

Their

use

for the

nonclassical

channel

is

entirely

equivalent

to that of

the

singlet

(1).

Maximal

entanglement

is

necessary

and

suKcient for

faithful

tele-

P)

will

be

reconstructed

(in

the

spin-2 case)

as a

ran-

dom mixture

of

the

four

states

of

Eq.

(6).

For

any

lg),

this

is a maximally

mixed

state,

giving

no information

about the

input

state

IP).

It

could

not

be

otherwise,

be-

cause

any

correlation between the

input

and the

guessed

output

could be used to

send

a superluminal

signal.

One

may

still

inquire

whether accurate

teleportation

of a

two-state

particle requires

a

full

two

bits of classical

information. Could

it

be

done,

for

example,

using

only

two or three

distinct

classical

messages

instead

of

four,

or

four

messages

of

unequal probability?

Later we

show

that a

full two

bits

of classical channel

capacity

are

neces-

sary.

Accurate teleportation

using

a

classical

channel of

any

lesser

capacity

would allow Bob to

send

superlumi-

nal

messages

through

the

teleported

particle,

by

guessing

the classical

message

before it

arrived

(cf.

Fig.

2).

Conversely

one

may

inquire

whether other states

be-

sides an EPR

singlet

can

be used

as

the nonclassical

chan-

nel of the teleportation process.

Clearly

any

direct

prod-

uct

state

of

particles

2

and

3

is

useless,

because

for such

states manipulation

of

particle

2 has

no effect

on what

can

be predicted

about particle

3.

Consider now a

non-

factorable state

IT2s)

. It can

readily

be

seen that

after

Alice's

measurement,

Bob's

particle

3

will be

related to

IP&)

by

four

fixed

unitary

operations

if

and

only

if

IT23)

has

the

form

Once Bob learns from Alice that

she

has obtained the

re-

sult

nm,

he

performs

on his

previously

entangled particle

(particle

3)

the

unitary

transformation

Two bits

Two

bits

U„=

)

e

'""~

A;)

((k+

m)

modNI.

k

EPR

pair

Two bits

EPR

pair

This transformation

brings

Bob

s

particle

to the origi-

nal

state of

Alice's

particle

1,

and the

teleportation

is

complete.

The classical

message

plays

an essential

role in telepor-

tation. To

see

why,

suppose

that

Bob

is

impatient,

and

tries

to

complete

the

teleportation

by

guessing

Alice's

classical

message

before it arrives.

Then

Alice's

expected

FIG.

1. Spacetime diagrams

for

(a)

quantum

teleporta-

tion,

and

(b)

4-way

coding

[12].

As

usual,

time increases

from bottom

to

top.

The

solid lines represent

a

classical

pair

of

bits,

the

dashed

lines an EPR

pair

of

particles (which

may

be

of

different

types),

and the

wavy

line a

quantum

parti-

cle

in

an

unknown

state

IP).

Alice

(A)

performs

a

quantum

measurement,

and

Bob

(B)

a

unitary

operation.

1897