Dense Quantum Coding and Quantum Finite Automata
ANDRIS AMBAINIS
Institute for Advanced Study, Princeton, New Jersey
ASHWIN NAYAK
California Institute of Technology, Pasadena, California
AMNON TA-SHMA
Tel-Aviv University, Tel-Aviv, Israel
AND
UMESH VAZIRANI
University of California, Berkeley, California
Abstract. We consider the possibility of encoding m classical bits into many fewer n quantum bits
(qubits) so that an arbitrary bit from the original m bits can be recovered with good probability.
We show that nontrivial quantum codes exist that have no classical counterparts. On the other hand,
we show that quantum encoding cannot save more than a logarithmic additive factor over the best
classical encoding. The proof is based on an entropy coalescence principle that is obtained by viewing
Holevo’s theorem from a new perspective.
In the existing implementations of quantum computing, qubits are a very expensive resource.
Moreover,itis difficult toreinitializeexistingbits during the computation.Inparticular,reinitialization
is impossible in NMR quantum computing, which is perhaps the most advanced implementation of
quantum computing at the moment. This motivates the study of quantum computation with restricted
memory and no reinitialization, that is, of quantum finite automata. It was known that there are
languages that are recognized by quantum finite automata with sizes exponentially smaller than those
of corresponding classical automata. Here, we apply our technique to show the surprising result that
there are languages for which quantum finite automata take exponentially more states than those of
corresponding classical automata.
Preliminary versions of this work appeared as Ambainis et al. [1999] and Nayak [1999b].
A. Ambainis was supported by the Berkeley Fellowship for Graduate Studies and, in part, by NSF
grant CCR-9800024; A. Nayak and U. Vazirani were supported by JSEP grant FDP 49620-97-1-
0220-03-98 and NSF grant CCR-9800024.
Authors’ addresses: A. Ambainis, Institute for Advanced Study, Einstein Dr., Princeton, NJ 08540,
e-mail: ambainis@ias.edu; A. Nayak, Computer Science Department and Institute for Quan-
tum Information, Mail Code 256-80, Pasadena, CA 91125, e-mail: nayak@cs.caltech.edu;
A. Ta-Shma, Computer Science Department, Tel-AvivUniversity, Ramat Aviv, Tel Aviv 69978, Israel,
e-mail: amnon@post.tau.ac.il; U. Vazirani, Computer Science Division, 671 Soda Hall, University of
California, Berkeley, CA, e-mail: vazirani@cs.berkeley.edu.
Permission to make digital/hard copy of part or all of this work for personal or classroom use is
granted without fee provided that the copies are not made or distributed for profit or commercial
advantage, the copyright notice, the title of the publication, and its date appear, and notice is given
that copying is by permission of ACM, Inc. To copy otherwise, to republish, to post on servers, or to
redistribute to lists requires prior specific permission and/or a fee.
C
°
2002 ACM 0004-5411/04/0700-0496 $5.00
Journal of the ACM, Vol. 49, No. 4, July 2002, pp. 496–511.
Dense Quantum Coding and Quantum Finite Automata 497
Categories and Subject Descriptors: F.2.0 [Analysis of Algorithms and Problem Complexity]:
General; F.1.1 [Computation by Abstract Devices]: Models of Computation—automata (e.g. finite,
push-down, resource-bounded)
General Terms: Theory
Additional Key Words and Phrases: Automaton size, communication complexity, encoding, finite
automata, quantum communication, quantum computation
1. Introduction
The tremendous information processing capabilities of quantum mechanical sys-
tems may be attributed to the fact that the state of an n quantum bit (qubit) system
is given by a unit vector in a 2
n
dimensional complex vector space. This suggests
the possibility that classical information might be encoded and transmitted with
exponentially fewer qubits. Yet, according to a fundamental result in quantum in-
formation theory, Holevo’s theorem [Holevo 1973], no more than n classical bits of
information can faithfully be transmitted by transferring n quantum bits from one
party to another. In view of this result, it is tempting to conclude that the exponen-
tially many degrees of freedom latent in the description of a quantum system must
necessarily stay hidden or inaccessible.
However, the situation is more subtle since the recipient of the n-qubit quan-
tum state has a choice of measurement he or she can make to extract information
about their state. In general, these measurements do not commute. Thus making a
particular measurement will disturb the system, thereby destroying some or all the
information that would have been revealed by another possible measurement. This
opens up the possibility of quantum random access codes, which encode classical
bits into many fewer qubits, such that the recipient can choose which bit of classical
information he or she would like to extract out of the encoding. We might think
of this as a disposable quantum phone book, where the contents of an entire tele-
phone directory are compressed into a few quantum bits such that the recipient of
these qubits can, via a suitably chosen measurement, look up any single telephone
number of his or her choice. Such quantum codes, if possible, would serve as a
powerful primitive in quantum communication.
Toformalize this, say we wish to encode m bits b
1
,...,b
m
into n qubits (m Àn).
Then a quantum random access encoding with parameters m, n, p (or simply
an m
p
7→ n encoding) consists of an encoding map from {0, 1}
m
to mixed states
with support in C
2
n
, together with a sequence of m possible measurements for the
recipient. The measurements are such that if the recipient chooses the ith measure-
ment and applies it to the encoding of b
1
···b
m
, the result of the measurement is b
i
with probability at least p.
The main point here is that since the m different possible measurements may
be noncommuting, the recipient cannot make the m measurements in succession
to recover all the encoded bits with a good chance of success. Thus the existence
of m
p
7→ n quantum random access codes with m Àn and p >
1
2
does not nec-
essarily violate Holevo’s bound. Furthermore, even though C
k
can accommodate
only k mutually orthogonal unit vectors, it can accommodate a
k
almost mutually
orthogonal unit vectors (i.e., vectors such that the inner product of any two has an
absolute value less than, say,
1
10
) for some a > 1. Indeed, there is no a priori reason
498 AMBAINIS ET AL.
to rule out the existence of codes that represent a
n
classical bits in n quantum bits
for some constant a > 1.
We start by showing that quantum encodings are more powerful than classi-
cal ones. We describe a
2
0.85
7→ 1
quantum encoding, and prove that there is no 2
p
7→ 1
classical encoding for any p >
1
2
. Our quantum encoding may be generalized to a
3
0.78
7→ 1
encoding, as was shown by Chuang [1997], and to encodings of more bits
into one quantum bit.
The main result in this paper is that (despite the potential of quantum encoding
shown by the arguments and results presented above) quantum encoding does not
provide much compression. We prove that any
m
p
7→ n
quantum encoding satis-
fies n ≥ (1 − H(p)) m, where H(p) =−plog p − (1 − p) log(1 − p)isthe
binary entropy function. The main technique in the proof is the use of the entropy
coalescence lemma, whichquantifies the increasein entropy when we take a convex
combination of mixed states. This lemma is obtained by viewing Holevo’s theorem
from a new perspective.
We turn to upper bounds on compression next, and show that the lower bound
is asymptotically tight up to an additive logarithmic term, and can be achieved
even with classical encoding. For any p > 1/2, we give a construction for m
p
7→ n
classical codes with n = (1 − H(p)) m + O(log m). Thus, even though quantum
random access codes can be more succinct as compared to classical codes, they
may be only a logarithmic number of qubits shorter.
In many of the existing quantum computing implementations, the complexity of
implementing the system grows tremendously as the number of qubits increases.
Moreover, even discarding one qubit and replacing it by a new qubit initialized
to
|
0
i
(often called a clean qubit) while keeping the total number of qubits the
same might be difficult or impossible (as in NMR quantum computing [Nielsen
and Chuang 2000]). This has motivated a huge body of work on one-way quantum
finite automata (QFAs), which are devices that model computers with a small finite
memory. During the computation of a QFA, no clean qubits are allowed, and in
addition no intermediate measurements are allowed, except to decide whether to
accept or reject the input.
We define generalized one-way quantum finite automata (GQFAs) that capture
the most general quantum computation that can be carried out with restricted mem-
oryand noextracleanqubits.In particular,themodel allowsarbitrarymeasurements
uponthestatespaceoftheautomaton as long as the measurements can be carried out
without clean qubits. We believe our model accurately incorporates the capabilities
of today’s implementations of quantum computing.
In Kondacs and Watrous [1997] it was shown that not every language recognized
by a classical deterministic finite automaton (DFA) is recognized by a QFA. On the
otherhand,therearelanguagesthatarerecognizedbyQFAswithsizesexponentially
smaller than those of corresponding classical automata [Ambainis and Freivalds
1998]. It remained open whether for any language that can be recognized by a
one-way finite automaton both classically and quantum-mechanically, a classical
automaton can be efficiently simulated by a QFA with no extra clean qubits. We
answer this question in the negative.
We apply the entropy coalescence lemma in a computational setting to give a
lower bound on the sizeof (GQFAs). We provethat there is a sequence of languages
for which the minimal GQFA has exponentially more states than the minimal DFA.
Dense Quantum Coding and Quantum Finite Automata 499
Itmaybesurprising that despite their quantum power(and irreversible computation,
thanks to the intermediate measurements) GQFAs are exponentially less powerful
for certain languages than classical DFAs. This lower bound highlights the need
for clean qubits for efficient computation.
2. Preliminaries
2.1. Q
UANTUM SYSTEMS. Just as a bit (an element of {0, 1}) is a fundamental
unitofclassicalinformation,aqubitisthefundamentalunitofquantuminformation.
A qubit is described by a unit vector in the two-dimensional Hilbert space C
2
. Let
|
0
i
and
|
1
i
be an orthonormal basis for this space.
1
In general, the state of the qubit
is a linear superposition of the form α
|
0
i
+β
|
1
i
. The state of n qubits is described
by a unit vector in the n-fold tensor product C
2
⊗C
2
⊗···⊗C
2
. An orthonormal
basis for this space is now given by the 2
n
vectors
|
x
i
, where x ∈{0,1}
n
. This is
often referred to as the computational basis. In general, the state of n qubits is a
linear superposition of the 2
n
computational basis states. Thus the description of
an n qubit system requires 2
n
complex numbers. This is arguably the source of the
astounding information processing capabilities of quantum computers.
The information in a set of qubits may be “read out” by measuring it in an
orthonormal basis, such as the computational basis. When a state
P
x
α
x
|
x
i
is
measured in the computational basis, we get the outcome x with probability
|
α
x
|
2
.
More generally, a (von Neumann) measurementon a Hilbert space H is definedby a
set of orthogonal projection operators
{
P
i
}
. When a state
|
φ
i
is measured according
to this set of projection operators, we get outcome i with probability
k
P
i
|
φ
ik
2
.
Moreover, the state of the qubits “collapses” to (i.e., becomes) P
i
|
φ
i
/
k
P
i
|
φ
ik
,
when the outcome i is observed. In order to retrieve information from an unknown
quantum state
|
φ
i
, it is sometimes advantageous to augment the state with some
ancillary qubits, so that the combined state is now
|
φ
i
⊗
¯
¯
¯
0
®
, before measuring
them jointly according to a set of operators
{
P
i
}
as above. This is the most general
form of quantum measurement, and is called a positive operator valued measure-
ment (POVM).
2.2. D
ENSITY MATRICES. In general, a quantum system may be in a mixed
state—a probability distribution over superpositions. For example, such a mixed
state may result from the measurement of a pure state
|
φ
i
.
Consider the mixed state {p
i
,
|
φ
i
i
}, where the superposition
|
φ
i
i
occurs with
probability p
i
. The behavior of this mixed state is completely characterized by its
density matrix ρ =
P
i
p
i
|
φ
i
ih
φ
i
|
. (The “bra” notation
h
φ
|
here is used to denote
the conjugate transpose of the superposition (column vector)
|
φ
i
. Thus
|
φ
ih
φ
|
denotes the outer product of the vector with itself.) For example, under a unitary
transformation U, the mixed state {p
i
,
|
φ
i
i
} evolves as
{
p
i
, U
|
φ
i
i}
, so that the
resulting density matrix is Uρ U
†
. When measured according to the projection
operators {P
j
}, the probability q
j
of getting outcome j is q
j
=
P
i
p
i
kP
j
|
φ
i
i
k
2
=
Tr(P
j
ρ P
j
), and the residual density matrix is P
j
ρ P
j
/q
j
. Thus, two mixed states
with the same density matrix have the same behavior under any physical operation.
We will therefore identify a mixed state with its density matrix.
1
This is Dirac’s ket notation.
|
φ
i
is another way of denoting a vector
E
φ.
500 AMBAINIS ET AL.
The following properties of density matrices follow from the definition. For any
density matrix ρ,
(1) ρ is Hermitian, that is, ρ = ρ
†
;
(2) ρ has unit trace, that is, Tr(ρ) =
P
i
ρ(i, i) = 1;
(3) ρ is positive semidefinite, that is,
h
ψ
|
ρ
|
ψ
i
≥ 0 for all
|
ψ
i
.
Thus, every density matrix is unitarily diagonalizable and has nonnegative real
eigenvalues that sum up to 1.
Recall that the amount of randomness (or the uncertainty) in a classical proba-
bility distribution may be quantified by its Shannon entropy. Doing the same for
a mixed state is tricky because all mixed states consistent with a given density
matrix are physically indistinguishable, and therefore contain the same amount of
“entropy.” Before we do this, we recall the classical definitions.
2.3. C
LASSICAL ENTROPY AND MUTUAL INFORMATION. The Shannon en-
tropy S(X) of a classical random variable X that takes values x in some finite
set with probability p
x
is defined as
S(X) =−
X
x
p
x
log p
x
.
The mutual information I(X : Y) of a pair of random variables X, Y is defined by
I(X : Y) =S(X)+ S(Y )− S(XY),
where XY denotes the joint random variable with marginals X and Y. It quantifies
the amount of correlation between the random variables X and Y.
Fano’s inequality asserts that if Y can predict X well, then X and Y have large
mutual information. We use a simple form of Fano’s inequality, referring only to
Boolean variables X and Y.
F
ACT 2.1(FANO’S INEQUALITY). Let X bea uniformly distributedboolean ran-
dom variable, and let Y be a boolean random variable such that Pr(X = Y) = p.
Then I(X : Y) ≥1 − H(p).
For other properties of these concepts we refer the reader to a standard text (such
as Cover and Thomas [1991]) on information theory.
2.4. V
ON NEUMANN ENTROPY. Consider the mixedstate X ={p
i
,
|
φ
i
i
}, where
the superposition
|
φ
i
i
occurs with probability p
i
. Since the constituent states
|
φ
i
i
of the mixture are not perfectly distinguishable in general, we cannot define the
entropy of this mixture to be the Shannon entropy of {p
i
}. Another way to see this
is that this mixture is equivalent to any other mixture with the same density matrix,
and so should have the same entropy as that mixture. Indeed, a special such equiv-
alent mixture can be obtained by diagonalizing the density matrix—the constituent
states of this mixture are orthogonal, and therefore perfectly distinguishable. Now,
the entropy of the density matrix can be defined to be the Shannon entropy of
these probabilities.
To formalize this, recall that every density matrix ρ is unitarily diagonalizable:
ρ =
X
j
λ
j
|ψ
j
ihψ
j
|,
