We describe a method to non-obliviously communicate a 2l-qubit quantum state
by physically transmitting l+o(l) qubits of communication, and by consuming l
ebits of entanglement and some shared random bits. In the non-oblivious
scenario, the sender has a classical description of the state to be
communicated. Our method can be used to communicate states that are pure or
entangled with the sender's system; l+o(l) and 3l+o(l) shared random bits are
sufficient respectively.Comment: 5 pages, revtex

A. Peres

A. S. Holevo

Aram Harrow

Debbie Leung

J. J. Duistermaat

Patrick Hayden

S. Lloyd

English

arXiv

We describe a method for nonobliviously communicating a 2l-qubit quantum state by physically transmitting l + o(l) qubits, and by consuming l ebits of entanglement plus some shared random bits. In the nonoblivious scenario, the sender has a classical description of the state to be communicated. Our method can be used to communicate states that are pure or entangled with the sender's system; l + o(l) and 3l + o(l) shared random bits are sufficient, respectively

Harrow, Aram

Hayden, Patrick

Leung, Debbie

Caltech Authors - Main

P H Y S I C A L R E V I E W L E T T E R S week ending7 MAY 2004VOLUME 92, NUMBER 18Superdense Coding of Quantum States
Aram Harrow
Physics Department, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
Patrick Hayden and Debbie Leung
Institute for Quantum Information, Caltech 107-81, Pasadena, California 91125, USA
(Received 30 August 2003; published 4 May 2004)187901-1We describe a method for nonobliviously communicating a 2l-qubit quantum state by physically
transmitting l ol qubits, and by consuming l ebits of entanglement plus some shared random bits. In
the nonoblivious scenario, the sender has a classical description of the state to be communicated. Our
method can be used to communicate states that are pure or entangled with the sender’s system; l ol
and 3l ol shared random bits are sufficient, respectively.
DOI: 10.1103/PhysRevLett.92.187901 PACS numbers: 03.67.Hk, 03.65.Taassumption about quantum communication, however, that
the sender (Alice) forwards or delivers a quantum state
without knowing what it is—a condition known as
trary 2l-qubit state can be prepared or shared with high
probability if Alice transmits l ol qubits to Bob and if
they share l ebits and at most Ol random classical bits.Introduction.—One of the most striking effects in
quantum information theory is known as superdense
coding [1]. By making use of shared entanglement, it is
possible to communicate classical information at twice
the rate one would naively expect is allowed by causality.
That is, by physically transmitting only one qubit (a two-
level system such as a spin-1=2 particle) while at the same
time consuming one ebit [the shared state j00i 
j11i= 2p ], it is possible to communicate two classical
bits worth of information. This observation can be sum-
marized by the following schematic inequality:
1 qubit  1 ebit  2 cbits:
It is natural to ask whether it is possible, using the same
resources, to communicate two qubits worth of quantum
information rather than just two classical bits. A simple
thought experiment reveals that this should not be the
case. Indeed, if the schematic inequality
1 qubit  1 ebit  2 qubits
were true, then the two qubits communicated using just a
qubit and an ebit could themselves be paired with two
ebits, resulting in the communication of four qubits worth
of quantum information. Repeating the process, an arbi-
trary amount of quantum information could be trans-
mitted by sending just the single original qubit and a
correspondingly large amount of entanglement. This is
known to be impossible [2]. For similar reasons, entan-
glement cannot increase the quantum capacity of a noise-
less quantum channel. This leads to a strong dichotomy:
entanglement can double the classical communication
capacity of a noiseless quantum channel but does not
increase the quantum communication capacity at all.
The argument for the latter claim rests on an important0031-9007=04=92(18)=187901(4)$22.50 ‘‘oblivious encoding.’’ Equivalently, Alice’s action is
required to be independent of the transmitted state.
Oblivious quantum communication automatically pre-
serves entanglement between the transmitted state and
any other system. In fact, recursive use of superdense
coding as described in the previous paragraph requires
that it preserve entanglement between the transmitted and
the receiver’s (Bob’s) systems.
Thus, the situation can be very different if Alice is
given a classical description of the state to be communi-
cated and can alter her encoding operation accordingly;
we call such encoding ‘‘nonoblivious.’’ An important
example of nonoblivious encoding is when Alice chooses
the communicated state herself, as is the case, for in-
stance, with quantum digital signatures [3]. Nonoblivious
encoding can be more powerful than the oblivious ver-
sion. This is true, for example, in remote state preparation
[4], the variant of teleportation [5] in which the sender
knows the state to be communicated. Remote state prepa-
ration requires only half of the communication resources
used in teleportation [6].
In this Letter we focus on nonoblivious communication
of quantum states using the dual resources of quantum
communication and entanglement. This is analogous to
remote state preparation but the classical communication
is now replaced by quantum communication. We restrict
our discussion to the tasks of ‘‘preparing’’ a pure state in
Bob’s system or ‘‘sharing’’ a pure state that is entangled
between Alice’s and Bob’s systems. (We avoid the term
‘‘transmitted state,’’ which can be confused with the
system that is physically transmitted.) Our goal is to
find out the extent to which entanglement can improve
the quantum communication capacity of a noiseless quan-
tum channel in the nonoblivious scenario.
Statement of result.—Our main result is that an arbi-2004 The American Physical Society 187901-1
P H Y S I C A L R E V I E W L E T T E R S week ending7 MAY 2004VOLUME 92, NUMBER 18No shared randomness is needed in the important cases of
communicating tensor product states or arbitrary states
drawn according to a known probability distribution. Our
result is a generalization of superdense coding to quan-
tum states in the asymptotic and nonoblivious scenario.
The possibility of superdense coding hinges on nonobli-
viousness. By Holevo’s theorem [2], both versions of
superdense coding use a minimal amount of communi-
cation, a rate at which the entanglement cost is also
optimal.
In the following, we first discuss the task of preparing
pure states, starting with a protocol that requires no
shared randomness but only succeeds with some poten-
tially small probability. We then describe a slight modifi-
cation that uses shared randomness to ensure a high
probability of success for all states. Finally, we describe
a generalization to share pure entangled states between
Alice and Bob.
An exact probabilistic protocol.—We first describe a
protocol for Alice to prepare any d2-dimensional state
j i in Bob’s system by sending logd qubits and consum-
ing logd ebits of shared entanglement. The protocol suc-
ceeds with a probability that depends on the state
prepared. Fix a basis fjiiAjjiBg1
i;j
d for Cd  Cd (or
equivalently Cd2). Alice and Bob initially share logd
ebits, or equivalently the maximally entangled state
jdi  1=

d
p Pdi1 jiiAjiiB. The state to be prepared,j i, can always be written as
j i  1
d
p X
i;j
xi;jjiiAjjiB  X  Ijdi; (1)
where X : Pi;jxi;jjiihjj. The identity
B : TrAj ih j  1dX
TX (2)
will be useful later. Equation (1) provides a simple scheme
to prepare the state j i—Alice applies X to her half of
jdi and then sends it to Bob:
In the above circuit and throughout this Letter, time goes
from left to right, single lines represent quantum regis-
ters, and registers connected in the left are initially in a
maximally entangled state. The arrow in Eq. (3) repre-
sents a register sent from Alice to Bob.
This scheme succeeds only if X is applied successfully,
which only occurs with some probability, because X may
not be unitary. One way to perform X is via a generalized
measurement [7] ! PkEkEyk with Kraus operators
E0  XkXk1 ; E1 

I  Ey0E0
q
; (4)187901-2where the operator norm of X, kXk1, can be taken to be
the square root of the largest eigenvalue of XyX. Then,
Ey0E0 
 I and the measurement is well defined. When the
measurement outcome is 0, X is successfully applied, and
this occurs with probability
T r Ey0E0
I
d
 Tr X
yX
dkXyXk1
 1
dkBk1 
1
1  : (5)
We have used Eq. (2) to obtain the equalities in Eq. (5).
The parameter  is defined by kBk1 
 1d 1 . It
measures the deviation of the state j i from being maxi-
mally entangled. This probability of success ranges from
1 for a maximally entangled state (  0) to 1=d for a
product state (  d 1). This implementation of X re-
quires the measurement outcome be sent to Bob, but it
suffices to use one extra qubit of communication, which is
negligible for large d.
High probability protocol for arbitrary pure states.—
If j i is chosen randomly, then with high probability it
will be highly entangled ( is small) [8–10] and thus we
would expect the protocol described above to succeed
with high probability.
Suppose our goal is to find a protocol that succeeds with
high probability for any choice of input state j i, includ-
ing product states. The above protocol can be adapted
easily if Alice and Bob share correlated random bits;
Alice will instead prepare Uj i, with U chosen according
to random bits shared with Bob. Bob can then undo U
after receiving Uj i. With high probability, the totally
random Uj i is highly entangled so that the probabilistic
protocol would succeed with high probability.
To make this intuition precise, we will analyze how
much randomness is required to ensure a given probabil-
ity of success. The answer is provided by a lemma that is
proved in the Appendix.
Lemma 1: Let 0<  
 1. If d  10 , there exists a set of
isometries fUkgnk1, where n  120 ln23 d logd, such that
8 j i; Prk

kTrAUkj ih jUyk k1 <
1 
d

 1 :
(6)
Here, each Uk takes d2-dimensional states into a Hilbert
space H A H B, where dimH A  112 ln22 d logd and
dimH B  d.
Our lemma states that, for any state j i, choosing Uk
randomly out of n  120 ln2
3
d logd possibilities will guar-
antee, with probability at least 1 , that the state Ukj i
to be prepared has kBk1 
 1d 1  and can be pre-
pared with probability at least 11 .
Thus, assuming the setting of Lemma 1, Eq. (6) is a
statement that the following protocol will succeed with
probability at least 11 for all states j i. To send any statej i: (1) Alice and Bob draw a random k 2 f1; . . . ; ng using
logn bits of shared randomness. Using Eq. (6), the proba-
bility that kTrAUkj ih jUyk k1 < 1d is at least 1 .187901-2
P H Y S I C A L R E V I E W L E T T E R S week ending7 MAY 2004VOLUME 92, NUMBER 18(2) Alice preparesUkj i using Eq. (3) and Bob appliesUyk
to obtain the correct state j i. With probability at least
1 , this procedure succeeds with probability greater
than 11 .
Let Ukj i  Xk  Ijdi. Note that Xk is a d n
matrix where n  112 ln2d logd=2. The entire proto-
col can be represented by the circuit
This protocol requires sending system A of dimension
112 ln2
2
d logd and consuming logd ebits. If the goal is to
prepare an arbitrary 2l-qubit state, we must take d  2l
and, upon taking logs, we find that the protocol requires
l logl 2 log1 7  l ol qubits of communica-
tion, l ebits, and logn  l logl 3 log1 7  l ol
shared random bits. Asymptotically, this gives a rate of 2
remote qubits prepared for every qubit communicated,
ebit consumed, and random bit shared.
Superdense coding of entangled states.—Our protocol
for preparing pure states can be easily adapted to enable
Alice and Bob to share a state j i starting from jdi. We
have an analog of the probabilistic protocol in Eq. (3):
where again j i  XA1A2  IBjdi. The nonunitary op-
eration X can be performed with probability
1=dkTrBj ih jk1.
Again, the idea is to randomize j i to make it highly
entangled across the A1A2 vs B partition. However, Bob
has no access to A1 so the randomization operation should
only act on A2 and B. The resulting protocol can be
represented by the circuit187901-3which is the analog of Eq. (7).
The above protocol shares a state with d2  22l dimen-
sions on Bob’s side. The quantum communication and
entanglement resources required are the same as in the
unentangled case, but more shared randomness is needed,
since the randomization is less effective when restricted
to part of the system. In fact, logn  3l 2 logl
5 log1 13 in this case. (We give a full proof of this result
in [11].)
Discussion.—The most intriguing open question is
whether shared randomness is required to perform super-
dense coding of quantum states. Our protocol is univer-
sal; the pure state to be prepared or shared is completely
arbitrary, and no restriction is imposed on its distribution.
Shared randomness is not needed when the state to be
communicated is drawn from an ensemble, and is negli-
gible when the state is a tensor product of arbitrary states
in blocks of ol qubits. We note that there is a completely
different method to achieve superdense coding of tensor
product quantum states (also without shared randomness)
using remote state preparation [6] and superdense coding
[1] in the framework of coherent classical communication
[13]. Conversely, teleportation [5] and superdense coding
of quantum states can be combined to rederive remote
state preparation [6], albeit with a new shared random-
ness cost.
A. H. acknowledges funding from the NSA and ARDA
under ARO Contract No. DAAD19-01-1-06. P. H. and
D. L. acknowledge the support of the Sherman Fair-
child, the Richard Tolman, and the Croucher foundations,
as well as the U.S. NSF under Grant No. EIA-0086038.
Appendix.—The Proof of Lemma 1 is as follows: Our
methods are similar to those detailed in [12]. Suppose
each Uk is drawn independently and identically distrib-
uted (i.i.d.) according to the Haar (i.e., left and right
unitarily invariant) measure [14]. First, fix two arbitrary
pure states in H B and H AB : H A H B and denote
their density matrices by  and  . Also, let dA 
dimH A and dB  dimH B. Then, for 0<  
 1,PrU

TrTrAU Uy  1dB 

2dB

 PrU

TrI U Uy  1
dB
 
2dB


 exp

 dA
2
14 ln2

;
where the inequality follows from an argument almost identical to the proof of Lemma II.3 in [12].
Our second step is to prove that
Pr U

kTrAU Uyk1  1dB 
3
4dB




10dB


2dB
exp

 dA
2
14 ln2

: (8)187901-3
P H Y S I C A L R E V I E W L E T T E R S week ending7 MAY 2004VOLUME 92, NUMBER 18M is called a -net for pure states in anm-dimensional Let MB be an  -net for pure states in H B. Then,
Hilbert space H if, for any pure state  2 H , 9 ~ 2
M such that k ~k1 
 . Here, k  k1 denotes the trace
norm, which is the sum of the absolute values of the
eigenvalues. Lemma II.4 in [12] states that 9 M such
that jMj 
 52m. We always refer to this type of net in
our discussion. Note that for any operator O 2 0; I,
Tr ~O 
 2 . We will use this fact often, and we
call it ‘‘fact 1.’’187901-42dB
using fact 1,
kTrU Uyk1  sup
2H B
TrTrAU Uy

 sup
~2MB
TrTrAU ~ Uy  4dB :
Thus,Pr U

kTrAU Uyk1  1dB
3
4dB


 PrU

sup
~2MB
Tr ~TrAU Uy  1dB

2dB




10dB


2dB
exp

dA2
14 ln 2

:; (9)
where the last line is obtained using the union bound.
In our third step, we introduce the binary random variables
Xk 

1 if kTrAUk Uyk k1  1dB 1 34 
0 otherwise:
(10)
Then the Xk are i.i.d. with expectation (over Uk) EXk 
  because of Eq. (9). For a fixed  ,
Pr fUkg

1
n
Xn
k1
Xk > 
	

 expnDk 
 exp

n



2dB log10dB 
dA
2
14 ln2

1
	

; (11)
where D is the divergence, with
Dk :  log 1  log1    log 1  log1  1  log:
Consider an 2dB -net MAB for j i 2 H A H B. Then
Pr fUkg

sup
~ 2MAB
1
n
Xn
k1
XK > 




10dB


2dAdB
exp

n



2dB log10dB 
dA
2
14 ln2

1
	

; (12)by the union bound. If
n >
2dAdB log10dB=
3dA=14 ln2  2dB log10dB=  1
; (13)
the probability in Eq. (12) is strictly less than 1 and there
exists a choice of fUkg such that the corresponding event
does not happen. That is,
sup
~ 2MAB
1
n
Xn
k1
XK 
 : (14)
Rephrasing the above using Eq. (10),
8 ~ 2MAB; Prk

kTrAUk ~ Uyk k1 
1
dB

1 3
4
	

 :
Finally, applying fact 1 to the net MAB,
8  2 H AB; 9 ~ such that
kTrAUk Uyk k1 
 kTrAUk ~ Uyk k1 

4dB
:
Putting the last two equations together,
8  2 H AB; Prk

kTrAUk Uyk k1 
1
dB
1 
	

 :
This completes the proof of Lemma 1 when we choose
dB  d and dA  112 ln22 d logd. With these parameters,
and with the hypothesis dB  10 , it is straightforward toverify that 120 ln2
3
d logd is an upper bound to the required
number of unitaries in Eq. (13).[1] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881
(1992).
[2] A. S. Holevo, Probl. Inf. Transm. 9, 177 (1973).
[3] D. Gottesman and I. L. Chuang, quant-ph/0105032.
[4] H.-K. Lo, Phys. Rev. A 62, 012313 (2000).
[5] C. H. Bennett, G. Brassard, C. Cre´peau, R. Jozsa,
A. Peres, and W. Wootters, Phys. Rev. Lett. 71, 1895
(1993).
[6] C. H. Bennett, P. Hayden, D.W. Leung, P.W. Shor, and
A. Winter, quant-ph/0307100.
[7] A. Peres, Quantum Theory: Concepts and Methods
(Kluwer, Dordrecht, 1995).
[8] S. Lloyd and H. Pagels, Ann. Phys. (N.Y.) 188 186, 1988.
[9] E. Lubkin, J. Math. Phys. (N.Y.) 19, 1028 (1978).
[10] D. Page, Phys. Rev. Lett. 71, 1291 (1993).
[11] A. Harrow, P. Hayden, and D.W. Leung, quant-ph/
0307221.
[12] P. Hayden, D.W. Leung, P.W. Shor, and A. Winter, quant-
ph/0307104.
[13] A. Harrow, Phys. Rev. Lett. 92, 097902 (2004).
[14] J. J. Duistermaat and J. A. C. Polk, Lie Groups (Springer-
Verlag, Berlin, 1999).187901-4


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