We give a transfer theorem for teleportation based on twisting the
entanglement measurement. This allows one to say what local unitary operation
must be performed to complete the teleportation in any situation, generalizing
the scheme to include overcomplete measurements, non-abelian groups of local
unitary operations (e.g., angular momentum teleportation), and the effect of
non-maximally entangled resources.Comment: 4 pages, 1 figur

A. Furasawa

C. H. Bennett

D. Boschi

D. Boumeester

D. Mayers

G. J. Milburn

G. M. D'Ariano

Giacomo M. D'Ariano

H.-K. Lo

L. Vaidman

M. A. Nielsen

Massimiliano F. Sacchi

S. L. Braunstein

Samuel L. Braunstein

English

arXiv

We give a transfer theorem for teleportation based on twisting the entanglement measurement. This allows one to say what local unitary operation must be performed to complete the teleportation in any situation, generalizing the scheme to include overcomplete measurements, non-abelian groups of local unitary operations (e.g., angular momentum teleportation), and the effect of non-maximally entangled resources

Braunstein, Samuel L.

D'Ariano, Giacomo M.

Milburn, Gerard J.

Sacchi, Massimiliano F.

University of Queensland eSpace

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Universal teleportation with a twist
Samuel L. Braunstein,1,2 Giacomo M. D’Ariano,3 G. J. Milburn4, and Massimiliano F. Sacchi3
1SEECS, University of Wales, Bangor LL57 1UT, UK
2Hewlett-Packard Labs, Mail Box M48, Bristol BS34 8QZ, UK
3INFM, Unita` di Pavia, Dipartimento di Fisica ’A. Volta’, Universita` di Pavia, via Bassi 6 – I-27100 Pavia, ITALY
4Department of Physics, The University of Queensland, QLD 4072 Australia
(July 7, 2002)
We give a transfer theorem for teleportation based on twisting the entanglement measurement. This
allows one to say what local unitary operation must be performed to complete the teleportation in
any situation, generalizing the scheme to include overcomplete measurements, non-abelian groups
of local unitary operations (e.g., angular momentum teleportation), and the effect of non-maximally
entangled resources.
PACS numbers: 03.65.Bz, 03.67.-a, 03.67.Hk
One of the most profound results of quantum infor-
mation theory is the discovery of quantum teleportation
protocols [1–4]. Teleportation is the disembodied trans-
port of quantum states between subsystems through a
classical communication channel requiring a shared re-
source of entanglement. The demonstration of teleporta-
tion elevates entanglement from a perennial theoretical
chestnut to a practical resource. The details of protocols
for teleportation may vary; specification of subsystems,
the shared entangled state, and the description of joint
measurements at the sender (Alice) or receiver (Bob).
For example already there have been several experimen-
tal implementations of teleportation [5–7] and other pro-
tocols have been proposed [8]. We show in this paper
that all teleportation schemes can be cast in a common
form with generalized (overcomplete) measurements and
which enables us to identify the local unitary operations
required to complete a teleportation scheme.
Let us start by recalling a maximally entangled state
in H⊗H
|Ψ〉〉 = 1√
d
∑
n
e−iφn |n〉 ⊗ |n〉 , (1)
where {|n〉} is any basis in H and d is the dimension of
H (the infinite dimensional case will be considered later).
In the following we will adopt the notation: use double
ket (bra) | · · · 〉〉 to denote vectors inH⊗H and customary
single ket (bra) for vectors in H.
It is now well known that we may span the set of all
maximally entangled states by local unitary operations.
Therefore, it is sufficient to consider local unitary oper-
ators acting only on one space. In what follows we shall
use a twist operation which swaps a pair of particles.
Here we introduce a ‘democratic’ notation so that given
a pair of systems in a state
|→Ψ〉〉 =
∑
jk
cjk|j〉 ⊗ |k〉 , (2)
the twisted/swapped version is denoted by
|←Ψ〉〉 =
∑
jk
cjk |k〉 ⊗ |j〉 , (3)
and vice-versa |←Ψ〉〉 ↔ |
→
Ψ〉〉 . The generalization to mixed
states follows trivially. A final piece of notation we in-
troduce is the transfer operator
Tba =
∑
n
|n〉b a〈n| . (4)
We now give several identities which succinctly de-
scribe teleportation. For an arbitrary maximally entan-
gled state |←Ψ〉〉 one can easily show that
〈〈→Ψ | ⊗ 1 1 ⊗ |
←
Ψ〉〉 = 1
d
T31 . (5)
Because this notation is rather compact, we shall rewrite
this equation labelling the particle numbers of each of
the states and operators involved, which gives
12〈〈
→
Ψ | ⊗ 1 3 1 1 ⊗ |
←
Ψ〉〉23 = 1
d
T31 . (6)
We feel that these equations are more pleasing and
equally unambiguous without particle labels [9].
Variations on the identity (5) are
〈〈→Ψ | ⊗ 1 |φ〉 ⊗ |
←
Ψ〉〉 = 1
d
|φ〉 , (7)
where |φ〉 is an arbitrary (unknown) quantum state; also
1 ⊗ 〈〈→Ψ | ⊗ 1 |Φ〉〉 ⊗ |
←
Ψ〉〉 = 1
d
|Φ〉〉 , (8)
and similarly
1 ⊗ 〈〈→Ψ | ⊗ 1 |
←
Ψ〉〉 ⊗ |Φ〉〉 = 1
d
|Φ〉〉 , (9)
which will correspond to entanglement swapping for
an arbitrary unknown (entangled) two-mode state |Φ〉〉.
Some other trivial variations of these identities are
1
〈〈←Ψ | ⊗ 1 1 ⊗ |
→
Ψ〉〉 = 1
d
T31 . (10)
and
1 ⊗ 〈〈→Ψ | |
←
Ψ〉〉 ⊗ 1 = 1 ⊗ 〈〈
←
Ψ | |
→
Ψ〉〉 ⊗ 1 = 1
d
T13 , (11)
and other identities analogous to Eqs. (7) and (8) follow.
Let us see how these identities allow us to understand
teleportation. Start with an unknown state and a shared
arbitrary maximally entangled resource |φ〉 ⊗ |Φ〉〉. Per-
form a measurement on the first two subsystems yielding
a maximally entangled result |→Ψ〉〉. We emphasize that
this measurement may be complete or overcomplete. In-
formation about which entangled state was found by Al-
ice is transmitted to Bob. To complete the teleportation
protocol Bob must convert |Φ〉〉 into the twisted version
of the entangled state actually found by Alice, i.e., |←Ψ〉〉.
This conversion involves a local unitary operation which
now leaves us with the situation described by Eq. (7).
Using it shows that the initial unknown state at Alice’s
end has been successfully transferred to Bob.
At this point it is worthwhile stepping back and look-
ing at what this teaches us about quantum teleportation.
In the ideal case Alice and Bob must share a maximally
entangled state and Alice must be able to perform a mea-
surement which yields a maximally entangled state. The
details of the measurement, for example, whether it in-
volves projection measurements or a POVM is unimpor-
tant. The reconstruction operation only relies on Bob be-
ing able to locally convert his shared entanglement into
the swapped version that Alice found. But this is a gen-
eral feature of maximally entangled states. In fact, it
lies at the heart of several other quantum communica-
tion protocols. In quantum dense coding this ability al-
lows us to encode the square as many orthogonal states
as are supported by the Hilbert space we are acting on
[10]. This yields potentially a doubled channel capacity.
Similarly, in any scheme which tries to implement bit
commitment, the freedom to locally convert any maxi-
mally entangled state to any other allows Alice to cheat
with impunity [11,12]. Now we have shown that this
same freedom also drives the quantum teleportation pro-
tocol. This commonality improves our understanding of
the ways in which the manipulation of shared entangle-
ment may be used.
In order to interpret the vector |→Ψ〉〉 as the result of
a measurement, we need an (over)complete set of maxi-
mally entangled vectors. This can be easily achieved by
having the unitary operator U ≡ U(g) as an element of
a group G = {g} of transformations g with unitary ir-
reducible representation (UIR) U(g) on Alice’s Hilbert
space H. Then, for any maximally entangled state |Ψ〉〉,
one has the identity∫
G
dg U(g)⊗ 1 |Ψ〉〉〈〈Ψ| U †(g)⊗ 1 = 1
d
1 ⊗ 1 , (12)
which easily follows from the identity (Schur’s lemma)
∫
G
dg U(g)AU †(g) = Tr(A) 1 , (13)
which holds for any operator A on H. The invariant
measure dg is normalized as
∫
G
dg |〈u|U(g)|v〉|2 = 1 , (14)
which is true for any pair of normalized vectors |u〉, and
|v〉 due to the irreducibility of the representation, (as-
suming square integrable UIR for simplicity). Eq. (12)
means that the set of vectors
|→Ψg〉〉 ≡ U(g)⊗ 1 |Ψ〉〉 , g ∈ G (15)
make a (generally not orthogonal) POVM that repre-
sents a measurement on H ⊗ H with result g. The
measurement correlates Alice’s Hilbert space with the
entangled resource. Alice gets the result g and com-
municates it to Bob classically, and as already men-
tioned Bob converts his shared entanglement |Φ〉〉 into
the twisted version of the entangled state found by Al-
ice, i.e., |←Ψg〉〉 = 1 ⊗U(g)|Ψ〉〉. For each result g the state
|φ〉 is teleported according to the overall transformation
〈〈→Ψg | ⊗ 1 |φ〉 ⊗ |
←
Ψg〉〉 = 1
d
|φ〉 , (16)
For discrete groups the sum replaces the integral over
G. Mathematically, Eq. (16) represents a pure instru-
ment [13], which describes the state reduction depending
on the outcome g of the measurement, and sends a pure
state into a pure state. In the general case such an in-
strument has the form
Ωx|φ〉
||Ωx|φ〉|| = |φx〉 , (17)
where x is the measurement outcome and |φx〉 is the state
conditioned by the result x. The case of teleportation is
peculiar because the conditioned state is identical to the
original one, independent of the measurement outcome,
and on the other hand it is “teleported” to another space.
In such a scenario the teleportation map should be re-
garded in the following way
Ωg |φ〉
||Ωg |φ〉|| = T31|φ〉 , (18)
where Ωg = 〈〈
→
Ψg | ⊗ 1 1 ⊗ |
←
Ψg〉〉 ≡ 1dT31. Notice that
Eq. (12) has the relevant feature that phase factors in the
group composition law can be neglected. In mathemati-
cal terms this means that if the unitary representation is
of the “projective” form
U(g)U(g′) = c(g, g′)U(gg′) , (19)
2
where c(g, g′) is a phase factor—a so called cocycle [14]—
then, because of the peculiar form of Eq. (12) the phase
factor c(g, g′) can be dropped.
The original case of Ref. [1] corresponds to the group
of the four Pauli matrices {1 , σx , σy , σz}, which is a pro-
jective representation of the abelian dihedral groupD2 of
pi-rotations around three perpendicular axes. Notice that
even though the projective representation is non-abelian
(i.e. σxσy = iσz = −σyσx) the represented group is
abelian (RxRy = RyRx = Rz , Rα denoting a pi-rotation
around the α = x, y, z axis).
The generalization to dimension N in Ref. [1] is again
a projective representation of an abelian group, namely
ZN × ZN , which is the group of discrete translations on
a lattice embedded in a torus. The representation of the
group given in Ref. [1] is
U(n,m) =
∑
k
e2piikn/N |k〉〈k ⊕m| , (20)
which satisfies the composition law
U(n,m)U(n′,m′) = e2piimn
′/NU(n⊕ n′,m⊕m′) , (21)
where n⊕ n′ denotes summation mod N .
For infinite dimensional Hilbert spaces the POVM re-
lated to Eq. (15) generally needs to be expressed in terms
of unnormalizable vectors. We rewrite Eq. (15) as follows
|→Θg〉〉 ≡ U(g)⊗ 1
∑
n
|n〉 ⊗ |n〉, g ∈ G (22)
(In general, the variable n may be continuous. In this
case the sum would be replaced by an integral.) More-
over, one needs to consider non-maximally entangled
states
|Ψ(λ) 〉〉 =
∑
n
cn(λ) |n〉 ⊗ |n〉 , (23)
which depend on a physical parameter λ ∈ [0, 1) (e.g.,
this could be a down-conversion gain) such that the state
becomes maximally entangled in the limit of λ→ 1 with
limλ→1 |cn+1(λ)/cn(λ)| = 1. Then we introduce the dis-
tortion operator
D(λ) =
∑
n
cn(λ) |n〉〈n| , (24)
and Eq. (16) now becomes
〈〈→Θg | ⊗ 1 1 ⊗ |
←
Ψg′ (λ) 〉〉 = U(g′)D(λ)U †(g) T31 , (25)
where
|←Ψg (λ)〉〉 = 1 ⊗ U(g) |Ψ(λ) 〉〉 . (26)
The teleportation map is achieved for g′ = g in the
limit of λ→ 1 as follows
lim
λ→1
U(g)D(λ)U †(g) T31|φ〉
||U(g)D(λ)U †(g) T31|φ〉|| = |φ〉 . (27)
The continuous variables teleportation of Ref. [3] is an
example of infinite dimensional teleportation. The group
is the Weyl-Heisenberg group of displacement operators
D(z) = eza
†−z¯a (where [a, a†] = 1 for the harmonic
oscillator algebra) with composition law D(z)D(w) =
eiIm(zw¯)D(z + w). Notice that this is just a projective
representation of the abelian group of translations on the
complex plane. Eq. (14) reads
∫
C
d2z
pi e
−|z|2 = 1 by tak-
ing |u〉 = |v〉 = |0〉 (|0〉 denoting the vacuum for a). The
entangled state is just the downconversion of the vacuum
|Ψ(λ) 〉〉 =
√
1− λ2
∞∑
n=0
λn|n〉 ⊗ |n〉 , (28)
(a phase factor for λ can always be included into the basis
definition). For λ < 1 one has the teleportation map with
distortion
Ω
(λ)
z |φ〉
||Ω(λ)z |φ〉||
= |φ(λ)z 〉 , (29)
where Ω
(λ)
z = 〈〈
→
Θz | ⊗ 1 1 ⊗ |
←
Ψz (λ) 〉〉, with |
←
Ψz (λ) 〉〉 =
1 ⊗ D(z)|Ψ(λ) 〉〉, and |→Θz 〉〉 = D(z) ⊗ 1 |Θ 〉〉, the latter
being the orthogonal POVM corresponding to the eigen-
vectors of the heterodyne photocurrent [15].
Teleportation for infinite dimensional Hilbert spaces is
not restricted to maximally entangled states based on
decomposition in Eq. (22). We can define teleportation
filters that only teleport part of the Hilbert space [17].
An example is the entangled state that results from two
harmonic oscillator coherent states |α〉⊗|β〉, through the
unitary transformation UK = exp(−ipia†ab†b) where a, b
are the annihilation operators. The resulting state is
|Π〉 = |α〉 ⊗ |β+〉+ | − α〉 ⊗ |β−〉 (30)
= |α+〉 ⊗ |β〉+ |α−〉 ⊗ | − β〉 (31)
where |z±〉 = |z〉±|−z〉 , which are sometimes called cat
states and are parity eigenstates (we have ignored nor-
malization). With this entangled resource/measurement
we can only teleport states that lie in the relevant two
dimensional parity subspace of the entire Hilbert space.
The universal scheme in the present letter allows tele-
portation through entangled measurements based on
non-abelian groups, which has never been considered yet.
The simplest case is angular momentum teleportation.
We parameterize the group representation matrices as
U(g) = exp(iϕ ~J · ~n), where ϕ ∈ [0, 2pi) [16], ~n is a unit
vector |~n|2 = 1 on a sphere, and Jα are customary angu-
lar momentum operators. With such a parameterization
the invariant measure is dg = d~n sin2(ϕ/2) dϕ/8pi. The
teleportation map is then
3
〈〈→Ψϕ,~n | ⊗ 1 |φ〉 ⊗ |
←
Ψϕ,~n〉〉 = 1
2J + 1
|φ〉 , (32)
where
|←Ψϕ,~n〉〉 ≡ 1 ⊗ eiϕ ~J·~n |Ψ〉〉 , (33)
for a fixed maximally entangled state |Ψ〉〉.
In this paper we have presented the essential mathe-
matical description of how entanglement plus local mea-
surement and unitary transformation enables teleporta-
tion. Dense coding [10] can be given a similar description,
however the role played by the classical and quantum in-
formation channels is interchanged (see figure 1). Both
schemes rely on the ability to map shared entanglement
to shared entanglement through local unitary transfor-
mations. We are thus able to see the common role of
local entanglement manipulation in quantum communi-
cation protocols.
[1] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A.
Peres, and W.K. Wooters, Phys. Rev. Lett. 70, 1895
(1993).
[2] L.Vaidman, Phys. Rev. A 49 1473 (1994).
[3] S.L. Braunstein and H.J. Kimble, Phys. Rev. Lett. 80,
869 (1998).
[4] D. Gottesman and I.L. Chuang, “Quantum teleportation
is a universal computation primitive,” LANL preprint,
quant-ph/9908010.
[5] D. Boumeester, J.-W. Pan, K. Mattle, M. Eibl, H. We-
infurter and A. Zeilinger, Nature (London) 390, 575
(1997).
[6] D. Boschi, S. Branca, F. De Martini, L. Hardy and S.
Popescu, Phys. Rev. Lett. 80, 1121 (1998).
[7] A. Furasawa, J.L. Sørensen, S.L. Braunstein, C.A. Fuchs,
H.J. Kimble and E.S. Polzik, Science 282, 706 (1998).
[8] G.J. Milburn and S.L. Braunstein, “Teleportation using
squeezed vacuum states”, to appear Phys. Rev. A (1999).
[9] The identity operators appearing in this expression
should be thought of as tensor slots with the first slot
for bras (linear functionals) in H3 and the second slot
for kets (vectors) in H1. When both slots are filled the
operator defines a linear functional on H1 ⊗H3 given by
3〈f |T31 |g〉1 = 3〈f |g〉3 = 1〈f |g〉1.
[10] C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett. 69, 2881
(1992).
[11] D. Mayers, Phys. Rev. Lett. 78, 3414 (1997).
[12] H.-K. Lo and H.F. Chau, Phys. Rev. Lett. 78, 3410
(1997).
[13] G.M. D’Ariano and H.P. Yuen, Phys. Rev. Lett. 76, 2832
(1996).
[14] The cocycle must satisfy the Jacobi associativity con-
straints, namely that c(gh, l) c(g, h) = c(g, hl) c(h, l) and
c(g, g−1) = c(g, e) = 1, for g, h, l ∈ G, e being the iden-
tity element.
FIG. 1. A schematic representation depicting how both
teleportation and dense coding use an entangled resource and
classical communication. Time runs vertically and space hor-
izontally. A single line represents a quantum state sent over
a quantum (noiseless) channel (Q), a double line represents
classical information sent over an ordinary classical communi-
cation channel (C). In dense coding the quantum and classical
channels are interchanged from that for teleportation. It is
already well known that the steps in each protocol converting
quantum to classical information (mediated by shared entan-
glement) involve common Bell state measurements. In this
paper, we have furthermore shown that those steps converting
classical to quantum information (mediated by shared entan-
glement) also operate on a common principle: one maximally
entangled state may be converted to any other by one-sided
(local) operations.
[15] G.M. D’Ariano and M.F. Sacchi, Phys. Rev. A 52, R4309
(1995).
[16] Due to the particular form of Eq. (12), the −1 factor of a
2pi rotation for half-odd spins is canceled between the bra
and ket. Therefore, it is sufficient to consider ψ ∈ [0, 2pi)
instead of ψ ∈ [0, 4pi).
[17] H.-K. Lo and H.F. Chau, Science 283, 2050 (1999).
4
AB
entangled resource
TELEPORTATION
A
B
entangled resource
DENSE CODING
Q
C
Q
Q
C
C
FIGURE 1


Universal Teleportation with a Twist

We give a transfer theorem for teleportation based on twisting the entanglement measurement, This allows one to say what local unitary operation must he performed to complete the teleportation in any situation, generalizing the scheme to include overcomplete measurements, non-Abelian groups of local unitary operations (e.g., angular momentum teleportation), and the effect of nonmaximally entangled resources

Braunstein, S. L.

DAriano, G. M.

Milburn, G. J.

Sacchi, M. F.

VOLUME 84, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 10 APRIL 2000
3486Universal Teleportation with a Twist
Samuel L. Braunstein,1,2 Giacomo M. D’Ariano,3 G. J. Milburn,4 and Massimiliano F. Sacchi3
1SEECS, University of Wales, Bangor LL57 1UT, United Kingdom
2Hewlett-Packard Labs, Mail Box M48, Bristol BS34 8QZ, United Kingdom
3INFM, Unità di Pavia, Dipartimento di Fisica “A. Volta,” Università di Pavia, via Bassi 6, I-27100 Pavia, Italy
4Department of Physics, The University of Queensland, Queensland 4072 Australia
(Received 12 August 1999)
We give a transfer theorem for teleportation based on twisting the entanglement measurement. This
allows one to say what local unitary operation must be performed to complete the teleportation in any
situation, generalizing the scheme to include overcomplete measurements, non-Abelian groups of local
unitary operations (e.g., angular momentum teleportation), and the effect of nonmaximally entangled
resources.
PACS numbers: 03.67.Hk, 03.65.BzOne of the most profound results of quantum informa-
tion theory is the discovery of quantum teleportation pro-
tocols [1–4]. Teleportation is the disembodied transport
of quantum states between subsystems through a classical
communication channel requiring a shared resource of en-
tanglement. The demonstration of teleportation elevates
entanglement from a perennial theoretical chestnut to a
practical resource. The details of protocols for teleporta-
tion may vary; specification of subsystems, the shared en-
tangled state, and the description of joint measurements at
the sender (Alice) or receiver (Bob). For example, already
there have been several experimental implementations of
teleportation [5–7] and other protocols have been proposed
[8]. We show in this paper that all teleportation schemes
can be cast in a common form with generalized (overcom-
plete) measurements and which enables us to identify the
local unitary operations required to complete a teleporta-
tion scheme.
Let us start by recalling a maximally entangled state in
H ≠ H
jC  1p
d
X
n
e2ifn jn ≠ jn , (1)
where jn is any basis in H and d is the dimension of
H (the infinite dimensional case will be considered later).
In the following we will adopt the notation: use double ket
(bra) j · · · to denote vectors in H ≠ H and customary
single ket (bra) for vectors in H .
It is now well known that we may span the set of all
maximally entangled states by local unitary operations.
Therefore, it is sufficient to consider local unitary operators
acting only on one space. In what follows we shall use a
twist operation which swaps a pair of particles. Here we
introduce a “democratic” notation so that given a pair of
systems in a state
j !C  
X
jk
cjkj j ≠ jk , (2)
the twisted/swapped version is denoted by0031-90070084(15)3486(4)$15.00j√C  
X
jk
cjkjk ≠ j j , (3)
and vice versa j√C  $ j !C . The generalization to mixed
states follows trivially. A final piece of notation we intro-
duce is the transfer operator [9]
Tba 
X
n
jnbanj . (4)
We now give several identities which succinctly describe
teleportation. For an arbitrary maximally entangled state
j√C  one can easily show that
1,2~~
!
C j√C 2,3  1d T31 , (5)
where the subscripts denote the particle numbers of each
of the states and operators involved.
Equation (5) implies
1,2~~
!
C j jf1 ≠ j√C 2,3  1d jf3 , (6)
where jf is an arbitrary (unknown) quantum state; also
for linearity
2,3~~
!
C j jF1,2 ≠ j√C 3,4  1d jF1,4 , (7)
and similarly
2,3~~
!
C j j√C 1,2 ≠ jF3,4  1d jF1,4 , (8)
which will correspond to entanglement swapping for an
arbitrary unknown (entangled) two-mode state jF. Some
other trivial variations of these identities are
1,2
√
C j !C 2,3  1d T31 , (9)
and
2,3
!
C j√C 1,2  2,3√C j j !C 1,2  1d T13 , (10)
and other identities analogous to Eqs. (6) and (7) follow.© 2000 The American Physical Society
VOLUME 84, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 10 APRIL 2000Let us see how these identities allow us to understand
teleportation. Start with an unknown state and a shared ar-
bitrary maximally entangled resource jf1 ≠ jF2,3. Per-
form a measurement on the first two subsystems yielding
a maximally entangled result j !C 1,2. We emphasize that
this measurement may be complete or overcomplete. In-
formation about which entangled state was found by Alice
is transmitted to Bob. To complete the teleportation proto-
col Bob must convert jF2,3 into the twisted version of the
entangled state actually found by Alice, i.e., j√C 2,3. This
conversion involves a local unitary operation which now
leaves us with the situation described by Eq. (6). Using
it shows that the initial unknown state at Alice’s end has
been successfully transferred to Bob.
At this point it is worthwhile stepping back and looking
at what this teaches us about quantum teleportation. In the
ideal case Alice and Bob must share a maximally entangled
state and Alice must be able to perform a measurement
which yields a maximally entangled state. The details of
the measurement, for example, whether it involves projec-
tion measurements or a positive operator valued measure
(POVM) is unimportant. The reconstruction operation re-
lies only on Bob being able to locally convert his shared
entanglement into the swapped version that Alice found.
But this is a general feature of maximally entangled states.
In fact, it lies at the heart of several other quantum com-
munication protocols. In quantum dense coding this ability
allows us to encode the square as many orthogonal states
as are supported by the Hilbert space we are acting on [10].
This yields potentially a doubled channel capacity. Simi-
larly, in any scheme which tries to implement bit com-
mitment, the freedom to locally convert any maximally
entangled state to any other allows Alice to cheat with
impunity [11,12]. Now we have shown that this same
freedom also drives the quantum teleportation protocol.
This commonality improves our understanding of the ways
in which the manipulation of shared entanglement may
be used.
In order to interpret the vector j !C  as the result of a
measurement, we need an (over)complete set of maximally
entangled vectors. This can be easily achieved by having
the unitary operator U  U	g
 as an element of a group
G  g of transformations g with unitary irreducible
representation (UIR) U	g
 on Alice’s Hilbert space H .
Then, for any maximally entangled state jC, one has the
identity
Z
G
dgU	g
 ≠ 'jC ~~CjUy	g
 ≠ '  1
d
' ≠ ' , (11)
which easily follows from the identity (Schur’s lemma)
Z
G
dgU	g
AUy	g
  Tr	A
' , (12)
which holds for any operator A on H . The invariant
measure dg is normalized asZ
G
dg jujU	g
 jyj2  1 , (13)
which is true for any pair of normalized vectors ju, and
jy due to the irreducibility of the representation (assum-
ing square integrable UIR for simplicity). Equation (11)
means that the set of vectors
j !Cg  U	g
 ≠ 'jC, g [ G (14)
makes a (generally not orthogonal) POVM that represents
a measurement on H ≠ H with result g. The measure-
ment correlates Alice’s Hilbert space with the entangled
resource. Alice gets the result g and communicates it to
Bob classically, and as already mentioned Bob converts
his shared entanglement jF2,3 into the twisted version
of the entangled state found by Alice, i.e., j√Cg2,3 
' ≠ U	g
 jC2,3. For each result g the state jf is tele-
ported according to the overall transformation
1,2~~
!
Cgj jf1 ≠ j√Cg2,3  1d jf3 . (15)
For discrete groups the sum replaces the integral over
G. Mathematically, Eq. (15) represents a pure instrument
[13], which describes the state reduction depending on the
outcome g of the measurement, and sends a pure state into
a pure state. In the general case such an instrument has
the form
Vxjf
kVxjfk  jfx , (16)
where x is the measurement outcome and jfx is the state
conditioned by the result x. The case of teleportation is
peculiar because the conditioned state is identical to the
original one, independent of the measurement outcome,
and on the other hand it is “teleported” to another space.
In such a scenario the teleportation map should be regarded
in the following way:
Vgjf1
kVgjf1k  T31jf1 , (17)
where Vg  1,2
!
Cg j√Cg2,3  1dT31. Notice that
Eq. (11) has the relevant feature that phase factors in the
group composition law can be neglected. In mathematical
terms this means that if the unitary representation is of the
“projective” form
U	g
U	g0
  c	g, g0
U	gg0
 , (18)
where c	g, g0
 is a phase factor—a so-called cocycle
[14]— then, because of the peculiar form of Eq. (11) the
phase factor c	g,g0
 can be dropped.
The original case of Ref. [1] corresponds to the group
of the four Pauli matrices ',sx ,sy ,sz, which is a pro-
jective representation of the Abelian dihedral group D2 of
p rotations around three perpendicular axes. Notice that
even though the projective representation is non-Abelian3487
VOLUME 84, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 10 APRIL 2000(i.e., sxsy  isz  2sysx), the represented group is
Abelian (RxRy  RyRx  Rz , Ra denoting a p rotation
around the a  x, y, z axis).
The generalization to dimension N in Ref. [1] is again
a projective representation of an Abelian group, namely,
N 3 N , which is the group of discrete translations on
a lattice embedded in a torus. The representation of the
group given in Ref. [1] is
U	n,m
 
X
k
e2piknN jk k © mj , (19)
which satisfies the composition law
U	n,m
U	n0,m0
  e2pimn
0NU	n © n0,m © m0
 , (20)
where n © n0 denotes summation modN .
For infinite dimensional Hilbert spaces the POVM re-
lated to Eq. (14) generally needs to be expressed in terms
of unnormalizable vectors. We rewrite Eq. (14) as follows:
j !Qg  U	g
 ≠ '
X
n
jn ≠ jn, g [ G . (21)
(In general, the variable n may be continuous. In this case
the sum would be replaced by an integral.) Moreover, one
needs to consider nonmaximally entangled states
jC	l
 
X
n
cn	l
 jn ≠ jn , (22)
which depend on a physical parameter l [ 0, 1
 (e.g., this
could be a down-conversion gain) such that the state be-
comes maximally entangled in the limit of l ! 1 with
liml!1 jcn11	l
cn	l
j  1. Then we introduce the dis-
tortion operator
D 	l
 
X
n
cn	l
 jn nj , (23)
and Eq. (15) now becomes
1,2~~
!
Qg j√Cg0	l
2,3  U	g0
D 	l
Uy	g
T31 , (24)
where
j√Cg	l
2,3  ' ≠ U	g
 jC	l
2,3 . (25)
The teleportation map is achieved for g0  g in the limit
of l ! 1 as follows:
lim
l!1
U	g
D 	l
Uy	g
T31jf1
kU	g
D 	l
Uy	g
T31jf1k  jf3 . (26)
The continuous variables teleportation of Ref. [3] is an
example of infinite dimensional teleportation. The group
is the Weyl-Heisenberg group of displacement operators
D	z
  ezay2z¯a (where a, ay  1 for the harmonic
oscillator algebra) with composition law D	z
D	w
 
ei Im	zw¯
D	z 1 w
. Notice that this is just a projective
representation of the Abelian group of translations on the
complex plane. Equation (13) reads
R

d2z
p e
2jzj2  1 by3488taking ju  jy  j0 (j0 denoting the vacuum for a).
The entangled state is just the down-conversion of the
vacuum
jC	l
 
p
1 2 l2
X`
n0
lnjn ≠ jn (27)
(a phase factor for l can always be included into the basis
definition). For l , 1 one has the teleportation map with
distortion
V	l
z jf1
kV	l
z jf1k
 jf	l
z 3 , (28)
where V	l
z  1,2~~
!
Qz j√Cz	l
2,,3, with j√Cz	l
2,3 
' ≠ D	z
 jC	l
2,3, and j !Qz1,2  D	z
 ≠ 'jQ1,2, the
latter being the orthogonal POVM corresponding to the
eigenvectors of the heterodyne photocurrent [15].
Teleportation for infinite dimensional Hilbert spaces
is not restricted to maximally entangled states based on
decomposition in Eq. (21). We can define teleportation
filters that only teleport part of the Hilbert space [16].
An example is the entangled state that results from two
harmonic oscillator coherent states ja ≠ jb, through the
unitary transformation UK  exp	2ipayabyb
 where
a, b are the annihilation operators. The resulting state is
jP  ja ≠ jb1 1 j 2 a ≠ jb2 (29)
 ja1 ≠ jb 1 ja2 ≠ j 2 b , (30)
where jz6  jz 6 j 2 z, which are sometimes called
cat states and are parity eigenstates (we have ignored nor-
malization). With this entangled resource/measurement
we can only teleport states that lie in the relevant two di-
mensional parity subspace of the entire Hilbert space.
The universal scheme in the present Letter allows
teleportation through entangled measurements based on
non-Abelian groups, which has never been considered
yet. The simplest case is angular momentum teleporta-
tion. We parametrize the group representation matrices as
U	g
  exp	iw J ? n
, wherew [ 0, 2p
 [17], n is a unit
vector j nj2  1 on a sphere, and Ja are customary angular
momentum operators. With such a parametrization the
invariant measure is dg  d n sin2	w2
dw8p . The
teleportation map is then
1,2
!
Cw, nj jf1 ≠ j√Cw, n2,3  12J 1 1 jf3 , (31)
where
j√Cw, n2,3  ' ≠ eiw J? njC2,3 , (32)
for a fixed maximally entangled state jC.
In this paper we have presented the essential mathemati-
cal description of how entanglement plus local measure-
ment and unitary transformation enables teleportation. In
this form we see that teleportation can be described as a
VOLUME 84, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 10 APRIL 2000FIG. 1. A schematic representation depicting how both tele-
portation and dense coding use an entangled resource and classi-
cal communication. Time runs vertically and space horizontally.
A single line represents a quantum state sent over a quantum
(noiseless) channel (Q), a double line represents classical infor-
mation sent over an ordinary classical communication channel
(C). In dense coding the quantum and classical channels are in-
terchanged from that for teleportation. It is already well known
that the steps in each protocol converting quantum to classical in-
formation (mediated by shared entanglement) involve common
Bell state measurements. In this paper, we have furthermore
shown that those steps converting classical to quantum informa-
tion (mediated by shared entanglement) also operate on a com-
mon principle: one maximally entangled state may be converted
to any other by one-sided (local) operations.
rather special POVM. Dense coding [10] can be given a
similar description, however the role played by the classi-
cal and quantum information channels is interchanged (seeFig. 1). Both schemes rely on the ability to map shared en-
tanglement to shared entanglement through local unitary
transformations. We are thus able to see the common role
of local entanglement manipulation in quantum communi-
cation protocols.
[1] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres,
and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
[2] L. Vaidman, Phys. Rev. A 49, 1473 (1994).
[3] S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett. 80, 869
(1998).
[4] D. Gottesman and I. L. Chuang, e-print quant-ph/9908010.
[5] D. Boumeester, J.-W. Pan, K. Mattle, M. Eibl,
H. Weinfurter, and A. Zeilinger, Nature (London) 390,
575 (1997).
[6] D. Boschi, S. Branca, F. De Martini, L. Hardy, and
S. Popescu, Phys. Rev. Lett. 80, 1121 (1998).
[7] A. Furasawa, J. L. Sørensen, S. L. Braunstein, C. A.
Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706
(1998).
[8] G. J. Milburn and S. L. Braunstein, Phys. Rev. A 60, 937
(1999).
[9] Notice that the transfer operator in Eq. (4) performs the
same transformation of the “swap” operator in M. A.
Nielsen and C. M. Caves, Phys. Rev. A 55, 2547 (1999).
[10] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881
(1992).
[11] D. Mayers, Phys. Rev. Lett. 78, 3414 (1997).
[12] H.-K. Lo and H. F. Chau, Phys. Rev. Lett. 78, 3410
(1997).
[13] G. M. D’Ariano and H. P. Yuen, Phys. Rev. Lett. 76, 2832
(1996).
[14] The cocycle must satisfy the Jacobi associativity con-
straints, namely, that c	gh, l
c	g,h
  c	g,hl
c	h, l
 and
c	g,g21
  c	g, e
  1, for g,h, l [ G, e being the iden-
tity element.
[15] G. M. D’Ariano and M. F. Sacchi, Phys. Rev. A 52, R4309
(1995).
[16] H.-K. Lo and H. F. Chau, Science 283, 2050 (1999).
[17] Because of the particular form of Eq. (11), the21 factor of
a 2p rotation for half-odd spins is canceled between the bra
and ket. Therefore, it is sufficient to consider c [ 0, 2p

instead of c [ 0, 4p
.3489


Universal teleportation with a twist

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Universal teleportation with a twist

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