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arXiv:quant-ph/0603135v1 15 Mar 2006
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Interaction in Quantum Communication
Hartmut Klauck, Ashwin Nayak, Amnon Ta-Shma and David Zuckerman
Hartmut is with the Department of Computer Science and Mathematics, University of Frankfurt, Robert Mayer
Strasse 11-15, 60054 Frankfurt am Main, Germany. His research is supported by DFG grant KL 1470/1. E-
mail: klauck@thi.informatik.uni-frankfurt.de. Most of this work was done while Hartmut was with the University
of Frankfurt, and later with CWI, supported by the EU 5th framework program QAIP IST-1999-11234 and
by NWO grant 612.055.001. Ashwin is with Department of Combinatorics and Optimization, and Institute for
Quantum Computing, University of Waterloo, 200 University Ave. W., Waterloo, ON N2L 3G1, Canada, E-mail:
anayak@math.uwaterloo.ca. He is also Associate Member, Perimeter Institute for Theoretical Physics, Canada.
Ashwin’s research is supported in part by NSERC, CIAR, MITACS, CFI, and OIT (Canada). Parts of this work
were done while Ashwin was at University of California, Berkeley, DIMACS Center and AT&T Labs, and California
Institute of Technology. Amnon is with the Dept. of Computer Science, Tel-Aviv University, Israel 69978, E-mail:
amnon@post.tau.ac.il. This research was supported in part by Grant No 2004390 from the United States-Israel
Binational Science Foundation (BSF), Jerusalem, Israel. A part of this work was done while Amnon was at the
University of California at Berkeley, and supported in part by a David and Lucile Packard Fellowship for Science
and Engineering and NSF NYI Grant CCR-9457799. David is with the Dept. of Computer Science, University
of Texas, Austin, TX 78712, E-mail: diz@cs.utexas.edu. This work was done while David was on leave at the
University of California at Berkeley. Supported in part by a David and Lucile Packard Fellowship for Science
and Engineering, NSF Grant CCR-9912428, NSF NYI Grant CCR-9457799, and an Alfred P. Sloan Research
Fellowship.
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Abstract
In some scenarios there are ways of conveying information with many fewer, even exponentially fewer,
qubits than possible classically [1], [2], [3]. Moreover, some of these methods have a very simple structure—
they involve only few message exchanges between the communicating parties. It is therefore natural to
ask whether every classical protocol may be transformed to a “simpler” quantum protocol—one that has
similar efficiency, but uses fewer message exchanges.
We show that for any constant k, there is a problem such that its k+1 message classical communication
complexity is exponentially smaller than its k message quantum communication complexity. This, in
particular, proves a round hierarchy theorem for quantum communication complexity, and implies, via
a simple reduction, an Ω(N1/k) lower bound for k message quantum protocols for Set Disjointness for
constant k.
Enroute, we prove information-theoretic lemmas, and define a related measure of correlation, the
informational distance, that we believe may be of significance in other contexts as well.
I. Introduction
A recurring theme in quantum information processing has been the idea of exploiting
the exponential resources afforded by quantum states to encode information in very non-
obvious ways. One representative result of this kind is due to Ambainis, Schulman, Ta-
Shma, Vazirani, and Wigderson [2]. They show that two players can deal a random set
√N cards each, from a pack of N cards, by the exchange of O(logN) quantum bits
of
between them. Another example is given by Raz [3] who shows that a natural geometric
promise problem that has an efficient quantum protocol, is hard to solve via classical
communication. Both are examples of problems for which exponentially fewer quantum
bits are required to accomplish a communication task, as compared to classical bits. A
third example is the O(√N logN) qubit protocol for Set Disjointness due to Buhrman,
Cleve, and Wigderson [1], which represents quadratic savings in the communication cost
over classical protocols.
The protocols presented by Ambainis et al. [2] and Raz [3] share the feature that they
require minimal interaction between the communicating players. For example, in the
protocol of Ambainis et al. [2] one player prepares a set of qubits in a certain state and
sends half of the qubits across as the message, after which both players measure their qubits
to obtain the result. In contrast, the protocol of Buhrman, Cleve and Wigderson [1] for
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checking set disjointness (DISJ) requires Ω(√N) messages. This raises a natural question:
Can we exploit the features of quantum communication and always reduce interaction
while maintaining the same communication cost? In particular, are there efficient quantum
protocols for DISJ that require only a few messages?
Kitaev and Watrous [4] show that every efficient quantum interactive proof can be trans-
formed into a protocol with only three messages of similar total length. This suggests that
it might be possible to reduce interaction in other protocols as well. In this paper we show
that for any constant k, there is a problem such that its k + 1 message classical commu-
nication complexity is exponentially smaller than its k message quantum communication
complexity, thus answering the above question in the negative. This, in particular, proves
a round hierarchy theorem for quantum communication complexity, and implies, via a
simple reduction, polynomial lower bounds for constant round quantum protocols for Set
Disjointness.
Our Separation Results
The role of interaction in classical communication is well-studied, especially in the con-
text of the Pointer Jumping function [5], [6], [7], [8], [9]. Our first result is for a subprob-
lem Skof Pointer Jumping that is singled out in Miltersen et al. [10] (see Section V-A for
a formal definition of Sk). We show:
Theorem I.1: For any constant k, there is a problem Sk+1such that any quantum pro-
tocol with only k messages and constant probability of error requires Ω(N1/(k+1)) commu-
nication qubits, whereas it can be solved with k + 1 messages by a deterministic protocol
with O(logN) bits.
A more precise version of this theorem is given in Section V-D and implies a round
hierarchy even when the number of messages k grows as a function of input size N, up
to k = Θ(logN/loglogN). Our analysis of Skfollows the same intuition as that behind
the result of Miltersen et al. [10], but relies on entirely new ideas from quantum information
theory. The resulting lower bound is optimal for a constant number of rounds.
Next, we study the Pointer Jumping function itself. Let fkdenote the Pointer Jumping
function with path length k + 1 on graphs with 2n vertices, as defined in Section VI.
The input length for the Pointer Jumping function fkis N = 2nlogn, independent of k,
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whereas the input length for Skis exponential in k. The function fkis thus usually more
appropriate for studying the effect of rounds on communication when k grows rapidly as
a function of the input length.
We first show an improved upper bound on the classical complexity of Pointer Jumping,
further closing the gap between the known classical upper and lower bounds. We then
turn into proving a quantum lower bound. We prove:
Theorem I.2: For any constant k, there is a classical deterministic protocol with k mes-
sage exchanges, that computes fkwith O(logn) bits of communication, while any k − 1
round quantum protocol with constant error for fkneeds Ω(n) qubits communication.
The lower bound of Theorem I.2 decays exponentially in k, and leads only to separation
results for k = O(logN). We believe it is possible to improve this dependence on k, but
leave it as an open problem. Note that in the preliminary version of this paper [11] this
decay was even doubly exponential, and the improvement here is obtained by using a
quantum version of the Hellinger distance.
Our lower bounds for Skand Pointer Jumping also have implications for Set Disjointness.
The problem of determining the quantum communication complexity of DISJ has inspired
much research in the last few years, yet the best known lower bound prior to this work
was Ω(logn) [2], [12]. We mentioned earlier the protocol of Buhrman et al. [1] which
solves DISJ with O(√N logN) qubits and Ω(√N) messages. Buhrman and de Wolf [12]
observed (based on a lower bound for random access codes [13], [14]) that any one message
quantum protocol for DISJ has linear communication complexity. We describe a simple
reduction from Pointer Jumping in a bounded number of rounds to DISJ and prove:
Corollary I.3: For any constant k, the communication complexity of any k-message
quantum protocol for Set Disjointness is Ω(N1/k).
A model of quantum communication complexity that has also been studied in the lit-
erature is that of communication with prior entanglement (see, e.g., Refs. [15], [12]). In
this model, the communicating parties may hold an arbitrary input-independent entangled
state in the beginning of a protocol. One can use superdense coding [16] to transmit n
classical bits of information using only ⌈n/2⌉ qubits when entanglement is allowed. The
players may also use measurements on EPR-pairs to create a shared classical random key.
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While the first idea often decreases the communication complexity by a factor of two, the
second sometimes saves logn bits of communication. It is unknown if shared entangle-
ment may sometimes decrease the communication more than that. Currently no general
methods for proving super-logarithmic lower bounds on the quantum communication com-
plexity with prior entanglement and unrestricted interaction are known. Our results all
hold in this model as well.
Our interest in the role of interaction in quantum communication also springs from the
need to better understand the ways in which we can access and manipulate information
encoded in quantum states. We develop information-theoretic techniques that expose
some of the limitations of quantum communication. We believe our information-theoretic
results are of independent interest.
The paper is organized as follows. In Section II we give some background on classical
and quantum information theory. We recommend Preskill’s lecture notes [17] or Nielsen
and Chuang’s book [18] as thorough introductions into the field. In Section III we present
new lower bounds on the quantum relative entropy function (Section III-A) and introduce
the informational distance (Section III-B). In Section IV we explain the communication
complexity model, followed by Section V where we prove our separation results and the
reduction to Set Disjointness (Section V-C). In Section VI we give our new upper bound
(Section VI-B) and quantum lower bound (Section VI-C) for the pointer-jumping problem.
Subsequent Results
Subsequent to the publication of the preliminary version of this paper [11] several new
related results have appeared.
communication complexity of the Set Disjointness problem is indeed Ω(√N), no matter
how many rounds are allowed. An upper bound of O(√N) is given by Aaronson and
First, Razborov proves in Ref. [19] that the quantum
Ambainis [20]. A result by Jain, Radhakrishnan, and Sen in Ref. [21] shows that the
complexity of protocols solving this problem in k rounds is at least Ω(n/k2). The same
authors show in Ref. [22] that quantum protocols with k−1 rounds for the Pointer Jumping
function fkhave complexity Ω(n/k4), but this result seems to hold only for the case of
protocols without prior entanglement. The same authors [23] also consider the complexity
of quantum protocols for the version of the Pointer Jumping function, in which not only
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one bit of the last vertex has to be computed, but its full name. Several papers ([24], [25],
[21], [22], [26]) have used the information theoretic techniques developed in the present
paper.
In this paper, we improve the dependence of communication complexity lower bounds
on the number of rounds, as compared to our results in Ref. [11]. To achieve this, we use a
different information-theoretic tool based on the quantum Hellinger distance. The version
of our Average Encoding Theorem based on Hellinger distance was independently found
by Jain et al. [21].
II. Information Theory Background
The quantum mechanical analogue of a random variable is a probability distribution
over superpositions, also called a mixed state. For the mixed state X = {pi,|φi?}, where
|φi? has probability pi, the density matrix is defined as ρX =
matrices are Hermitian, positive semi-definite, and have trace 1. I.e., a density matrix has
?
ipi|φi??φi|. Density
an eigenvector basis, all the eigenvalues are real and between zero and one, and they sum
up to one.
A. Trace Norm And Fidelity
The trace norm of a matrix A is defined as ?A?t= Tr√A†A, which is the sum of the
magnitudes of the singular values of A. Note that if ρ is a density matrix, then it has
trace norm one. If φ1,φ2are pure states then:
?|φ1??φ1| − |φ2??φ2|?t
=2
?
1 − |?φ1|φ2?|2.
We will need the following consequence of Kraus representation theorem (see for example
Preskill’s lecture notes [17]):
Lemma II.1: For each Hermitian matrix ρ and each trace-preserving completely positive
superoperator T: ?T(ρ)?t≤ ?ρ?t.
A useful alternative to the trace metric as a measure of closeness of density matrices is
fidelity. Let ρ be a mixed state with support in a Hilbert space H. A purification of ρ is
any pure state |φ? in an extended Hilbert space H ⊗ K such that TrK|φ??φ| = ρ. Given
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two density matrices ρ1,ρ2on the same Hilbert space H, their fidelity is defined as
F(ρ1,ρ2) = sup |?φ1|φ2?|2,
where the supremum is taken over all purifications |φi? of ρiin the same Hilbert space.
Jozsa [27] gave a simple proof, for the finite dimensional case, of the following remarkable
equivalence first established by Uhlmann [28].
Fact II.2 (Jozsa) For any two density matrices ρ1,ρ2 on the same finite dimensional
space H,
F(ρ1,ρ2) =
?
Tr
??
ρ11/2ρ2ρ11/2??2
= ?√ρ1√ρ2?2
t.
Using this equivalence, Fuchs and van de Graaf [29] relate fidelity to the trace distance.
Fact II.3 (Fuchs, van de Graaf) For any two mixed states ρ1,ρ2,
1 −
?
F(ρ1,ρ2) ≤
1
2?ρ1− ρ2?t
≤
?
1 − F(ρ1,ρ2).
While the definition of fidelity uses purifications of the mixed states and relates them
via the inner product, fidelity can also be characterized via measurements (see Nielsen and
Chuang [18]).
Fact II.4: For two probability distributions p,q on finite sample spaces, let F(p,q) =
√piqi)2denote their fidelity. Then, for any two mixed states ρ1,ρ2,
(?
i
F(ρ1,ρ2)= min
{Em}F(pm,qm),
where the minimum is over all POVMs {Em}, and pm= Tr(ρ1Em),qm= Tr(ρ2Em) are
the probability distributions created by the measurement on the states.
A useful property of the trace distance ?ρ1− ρ2?tas a measure of distinguishability is
that it is a metric, and hence satisfies the triangle inequality. This is not true for fidelity
F(ρ1,ρ2) or for 1−F(ρ,ρ2). Fortunately, a variant of fidelity is actually a metric. Denote
by
?
the quantum Hellinger distance. Clearly h(ρ1,ρ2) inherits most of the desirable properties
h(ρ1,ρ2)=1 −
?
F(ρ1,ρ2)
of fidelity, like unitary invariance, definability as a maximum over all measurements of the
classical Hellinger distance of the resulting distributions, and so on. To see that h(ρ1,ρ2)
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is actually a metric one can simply use Fact II.4 to reduce this problem to showing that
the classical Hellinger distance is a metric, which is well known.
Analogously to Lemma II.1, due to the monotonicity of fidelity [18], we have:
Lemma II.5: For all density matrices ρ1,ρ2and each trace-preserving completely posi-
tive superoperator T: h(T(ρ1),T(ρ2)) ≤ h(ρ1,ρ2).
Let us also note the following relation between the Hellinger distance and the trace
norm that follows directly from Fact II.3.
Lemma II.6: For any two mixed states ρ1,ρ2,
h2(ρ1,ρ2) ≤
1
2?ρ1− ρ2?t
≤
√2 · h(ρ1,ρ2).
We will sometimes work with h2(·,·) instead of h(·,·). This is not a metric, but it is
true that for all density matrices ρ1,ρ2,ρ3:
h2(ρ1,ρ2) ≤ (h(ρ1,ρ3) + h(ρ3,ρ2))2≤ 2h2(ρ1,ρ3) + 2h2(ρ3,ρ2).
B. Local Transition Between Bipartite States
Jozsa [27] proved:
Theorem II.7 (Jozsa) Suppose |φ1?,|φ2? ∈ H ⊗ K are the purifications of two density
matrices ρ1,ρ2 in H. Then, there is a local unitary transformation U on K such that
F(ρ1,ρ2) = |?φ1|(I ⊗ U)|φ2?|2.
As noticed by Lo and Chau [30] and Mayers [31], Theorem II.7 immediately implies
that if two states have close reduced density matrices, than there exists a local unitary
transformation transforming one state close to the other. Formally,
Lemma II.8: (Local Transition Lemma, based on Refs. [30], [31], [27], [29]) Let ρ1,ρ2
be two mixed states with support in a Hilbert space H. Let K be any Hilbert space of
dimension at least dim(H), and |φi? any purifications of ρiin H ⊗ K.
Then, there is a local unitary transformation U on K that maps |φ2? to |φ′
such that
2? = I⊗U |φ2?
h(|φ1??φ1|,|φ′
2??φ′
2|) = h(ρ1,ρ2).
Furthermore,
?|φ1??φ1| − |φ′
2??φ′
2|?t
≤ 2?ρ1− ρ2?
1
2
t.
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Proof: (Of Lemma II.8): By Theorem II.7, there is a (local) unitary transformation U
on K such that (I ⊗U)|φ2? = |φ′
Hence the statement about the Hellinger distance holds.
2?, a state which achieves fidelity: F(ρ1,ρ2) = |?φ1|φ′
2?|2.
By Lemma II.6
?|φ1??φ1| − |φ′
≤ 2√2 · h(|φ1??φ1|,|φ′
= 2√2 · h(ρ1,ρ2)
≤ 2 · ?ρ1− ρ2?
2??φ′
2|?t
2??φ′
2|)
1
2
t.
C. Entropy, Mutual Information, And Relative Entropy.
H(·) denotes the binary entropy function H(p) = plog(1
non entropy S(X) of a classical random variable X on a finite sample space is?
where px is the probability the random variable X takes value x. The mutual infor-
p)+(1−p)log(
1
1−p). The Shan-
xpxlog(1
px)
mation I(X : Y ) of a pair of random variables X,Y is defined to be I(X : Y ) =
H(X) + H(Y ) − H(X,Y ). For other equivalent definitions, and more background on
the subject see, e.g., the book by Cover and Thomas [32].
We use a simple form of Fano’s inequality.
Fact II.9 (Fano’s inequality) Let X be a uniformly distributed Boolean random vari-
able, and let Y be a Boolean random variable such that Prob(X = Y ) = p. Then I(X :
Y ) ≥ 1 − H(p).
The Shannon entropy and the mutual information functions have natural generalizations
to the quantum setting. The von Neumann entropy S(ρ) of a density matrix ρ is defined
as S(ρ) = −Trρlogρ = −?
of ρ. Notice that the eigenvalues of a density matrix form a probability distribution. In
iλilogλi, where {λi} is the multi-set of all the eigenvalues
fact, we can think of the density matrix as a mixed state that takes the i’th eigenvector
with probability λi. The von Neumann entropy of a density matrix ρ is, thus, the entropy
of the classical distribution ρ defines over its eigenstates.
The mutual information I(X : Y ) of two disjoint quantum systems X,Y is defined to
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be I(X : Y ) = S(X)+S(Y )−S(XY ), where XY is the density matrix of the system that
includes the qubits of both systems. Then
I(X : Y Z) = I(X : Y ) + I(XY : Z) − I(Y : Z),
I(X : Y Z) ≥ I(X : Y ),
(1)
(2)
Equation (2) is in fact equivalent to the strong sub-additivity property of von Neumann
entropy.
We need the following slight generalization of Theorem 2 in Cleve et al. [15].
Lemma II.10: Let Alice own a state ρAof a register A. Assume Alice and Bob com-
municate and apply local transformations, and at the end register A is measured in the
standard basis. Assume Alice sends Bob at most k qubits, and Bob sends Alice arbitrarily
many qubits. Further assume all these local transformations do not change the state of
register A, if A is in a classical state. Let ρABbe the final state of A and Bob’s private
qubits B. Then I(A : B) ≤ 2k.
Proof: Considering the joint state of register A and Bob’s qubits, there cannot be any
interference between basis states differing on A. Thus we can assume that ρAis measured
in the beginning, i.e., that ρA is classical. In this case the result directly follows from
Theorem 2 in Ref. [15].
Note that in the above lemma Alice and Bob can use Bob’s free communication to set
up an arbitrarily large amount of entanglement independent of ρA.
The relative von Neumann entropy of two density matrices, defined by S(ρ?σ) =
Trρlogρ − Trρlogσ. One useful fact to know about the relative entropy function is
that I(A : B) = S(ρAB?ρA⊗ρB). For more properties of this function see Refs. [17], [18].
III. Informational Distance And New Lower Bounds On Relative
Entropy
A. New Lower Bounds On Relative Entropy
We now prove that the relative entropy S(ρ1?ρ2) is lower bounded by Ω(?ρ1− ρ2?2
and by Ω(h2(ρ1,ρ2)). We believe these results are of independent interest. A classical
t)
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version of the theorem can be found in, e.g., Cover and Thomas’ book on Information
Theory [32].
Theorem III.1: For all density matrices ρ1,ρ2:
S(ρ1?ρ2)≥
1
2ln2?ρ1− ρ2?2
t.
Although this relationship has appeared in the literature [33], it was rediscovered by
several authors, including us. Below we give a proof of this theorem for completeness. The
earlier version of our paper [11] contained a more complicated proof.
Proof: (Theorem III.1) The proof goes by reduction to the classical case. Consider
the classical distributions ˜ ρ1, ˜ ρ2 obtained by measuring ρ1,ρ2in the basis diagonalizing
their difference ρ1− ρ2. It is known [17], [18] that
? ˜ ρ1− ˜ ρ2?1
= ?ρ1− ρ2?t.
Due to Lindblad-Uhlmann monotonicity of relative von Neumann entropy [17], [18],
S(ρ1?ρ2) ≥ S(˜ ρ1?˜ ρ2).
The classical version of the theorem [32] now gives
S(˜ ρ1?˜ ρ2) ≥
1
2ln2? ˜ ρ1− ˜ ρ2?2
1
2ln2?ρ1− ρ2?2
1
=
t.
This completes the proof.
Now we show an analogous result for the quantum Hellinger distance.
Theorem III.2: For all density matrices ρ1,ρ2:
S(ρ1?ρ2)≥
2
ln2h2(ρ1,ρ2).
This theorem has also been shown independently by Jain et al. [21].
Proof:
We first show that the theorem holds when ρ1and ρ2are classical distribu-
tions, and then generalize this to the quantum case.
In the classical case we first show S(ρ1?ρ2) ≥ −2log(1 − h2(ρ1,ρ2)). This was shown
by Dacunha-Castelle in Ref. [34].
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log(1 − h2(ρ1,ρ2)) = log(
?
??
??
F(ρ1,ρ2))
= log
i
?
ρ1(i)ρ2(i)
?
?
?
= log
i
ρ1(i)
?ρ2(i)
?ρ1(i)
??ρ2(i)
?ρ1(i)
≥
?
i
ρ1(i)log
= −1
2S(ρ1?ρ2).
The first equation is by definition of h, the second by definition of the classical fidelity
function, and the inequality is by an application of Jensen’s inequality.
Having that, S(ρ1?ρ2) ≥
the theorem holds in the classical case.
2
ln2h2(ρ1,ρ2) using −ln(1 − x) ≥ x for all 0 ≤ x ≤ 1 and so
To show the quantum case recall that both h(·,·) and S(·?·) can be defined as the max-
imum over all POVM measurements of the classical versions of these functions on the dis-
tributions obtained by the measurements. Fix a POVM {Em} that maximizes h(p,q) for
the distributions p,q obtained from ρ1,ρ2. Then S(ρ1?ρ2) ≥ S(p?q) by Lindblad-Uhlmann
monotonicity, and S(p?q) ≥
result follows.
2
ln2h2(p,q) =
2
ln2h2(ρ1,ρ2) because h(p,q) = h(ρ1,ρ2). The
B. Informational Distance
From Theorem III.2 follows that for a bipartite state ρAB,
I(A : B)=S(ρAB?ρA⊗ ρB)≥
2
ln2h2(ρAB,ρA⊗ ρB).
Thus the distance between the tensor product state and the “real” (possibly entangled)
bipartite state can be bounded in terms of the Hellinger distance. We call the quantity
D(A : B) = h(ρAB,ρA⊗ ρB) the “informational distance.”
amount of correlation between the quantum registers A and B, and can be positive even
D(A : B) measures the
when the system is classical or not entangled. Later we state some of its properties and use
it for proving the quantum communication lower bound on the pointer jumping problem.
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The next lemma collects a few immediate properties of informational distance.
Lemma III.3: For all states ρXY Zthe following hold:
1. D(X : Y ) = D(Y : X),
2. 0 ≤ D(X : Y ) ≤ 1,
3. D(X : Y ) ≥ h(T(ρXY),T(ρX⊗ ρY)) for all completely positive, trace-preserving su-
peroperators T,
4. D(XY : Z) ≥ D(X : Z),
5. D(X : Y ) ≤?I(X : Y ).
Proof:
(1) is true by definition, (2) follows from the definition and the triangle
inequality, (3,4) follow from Lemma II.5 and (5) from Theorem III.2.
We now examine the informational distance in the special case where ρQX is block
diagonal, with classical ρX. We denote by ρ(x)
Qthe density matrix obtained by fixing X to
some classical value x and normalizing. Pr(x) is the probability of X = x.
Lemma III.4: For all block diagonal ρQX, where ρXcorresponds to a classical distribu-
tion,
1. D2(Q : X) = Exh2?
2. Further assume X is Boolean with Pr(X = 1) = Pr(X = 0) = 1/2. Let there be a
ρ(x)
Q,ρQ
?
.
measurement acting on the Q system only, yielding a Boolean random variable Y with
Pr(X = Y ) ≥ 1 − ǫ and Pr(X ?= Y ) ≤ ǫ. Then D2(Q : X) ≥ 1/8 − ǫ/2.
The first item is true because ρQXis block-diagonal with respect to X. In the second item,
notice that the same measurement applied to ρX⊗ ρQyields a distribution with Pr(X =
Y ) = Pr(X ?= Y ) = 1/2, because Q is independent of X, and X is uniform. Observe
that ?ρXQ− ρX⊗ ρQ?t≥ ?ρXY− ρX⊗ ρY?t≥ 1−2ǫ and then apply Lemma II.6. Note
that this is a rather crude estimate, since D(Q : X) approaches 1 − 1/√2 when ǫ goes to
zero.
C. The Average Encoding Theorem
A corollary of Theorems III.1,III.2 is the following “Average encoding theorem”:
Theorem III.5 (Average encoding theorem) Let x ?→ ρxbe a quantum encoding map-
ping an m bit string x ∈ {0,1}minto a mixed state with density matrix ρx. Let X be
distributed over {0,1}m, where x ∈ {0,1}mhas probability px, let Q be the encoding of X
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according to this map, and let ¯ ρ =?
xpxρx. Then,
?
x
px? ¯ ρ − ρx?t
≤ [(2ln2) I(Q : X)]1/2
and
?
x
px h2(¯ ρ,ρx) ≤
ln2
2
I(Q : X).
In other words, if an encoding Q is only weakly correlated to a random variable X, then
the “average encoding” ¯ ρ is in expectation (over a random string) a good approximation
of any encoded state. Thus, in certain situations, we may dispense with the encoding
altogether, and use the single state ¯ ρ instead. The preliminary version of our paper [11]
did not include the second statement. The present stronger version was also observed
independently by Jain et al. [21].
Proof: (Of Theorem III.5) In the setting of the Average encoding theorem we have
a random variable that is distributed over {0,1}m, and a quantum encoding x ?→ ρx
mapping m bit strings x ∈ {0,1}minto mixed states with density matrices ρx. Let X be
the register holding the input x and Q be the register holding the encoding. Let us also
define the average encoding ¯ ρ =?
Then, by Theorem III.1,
xpxρx.
I(Q : X) = S(ρQX?ρQ⊗ ρX)≥
1
2ln2?ρQX− ρQ⊗ ρX?2
t
The density matrix ρX of the X register alone is diagonal and contains the values
pxon the diagonal, the density matrix ρQof the Q register alone is ¯ ρ, and the density
matrix ρQ⊗ ρXis block diagonal and the x’th block is of the form px¯ ρ. Also, the density
matrix ρQX of the whole system is block diagonal, with pxρx in the x’th block. Thus,
xpx?ρx− ¯ ρ?t, and so Ex?ρx− ¯ ρ?t≤√2ln2?I(Q : X).
The second statement follows analogously using Theorem III.2.
?ρQX− ρQ⊗ ρX?t=?
IV. The Communication Complexity Model
In the quantum communication complexity model [35], two parties Alice and Bob hold
qubits. When the game starts Alice holds a classical input x and Bob holds y, and so
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the initial joint state is simply |x? ⊗ |y?. Furthermore each player has an arbitrarily large
supply of private qubits in some fixed basis state. The two parties then play in turns.
Suppose it is Alice’s turn to play. Alice can do an arbitrary unitary transformation on
her qubits and then send one or more qubits to Bob. Sending qubits does not change
the overall superposition, but rather changes the ownership of the qubits, allowing Bob
to apply his next unitary transformation on the newly received qubits. Alice may also
(partially) measure her qubits during her turn. At the end of the protocol, one player
makes a measurement and declares the result of the protocol. In a classical probabilistic
protocol the players may only exchange classical messages.
In both the classical and quantum settings we can also define a public coin model.
In the classical public coin model the players are also allowed to access a shared source
of random bits without any communication cost. The classical public and private coin
models are strongly related [36]. Similarly, in the quantum public coin model Alice and
Bob initially share an arbitrary number of quantum bits which are in some pure state
that is independent of the inputs. This is better known as communication with prior
entanglement [15], [12].
The complexity of a quantum (or classical) protocol is the number of qubits (respectively,
bits) exchanged between the two players. We say a protocol computes a function f :
X × Y ?→ {0,1} with ǫ ≥ 0 error if, for any input x ∈ X,y ∈ Y, the probability that the
two players compute f(x,y) is at least 1 − ǫ. Qǫ(f) (resp. Rǫ(f)) denotes the complexity
of the best quantum (resp. probabilistic) protocol that computes f with at most ǫ error.
For a player P ∈ {Alice, Bob}, Qc,P
protocol that computes f with at most ǫ error with only c messages (called rounds in the
ǫ (f) denotes the complexity of the best quantum
literature), where the first message is sent by P. If the name of the player is omitted
from the superscript, either player is allowed to start the protocol. We say a protocol P
computes f with ǫ error with respect to a distribution µ on X × Y, if
Prob(x,y)∈µ,P(P(x,y) = f(x,y)) ≥ 1 − ǫ.
Qc,P
only c messages where the first message is sent by player P. We will use the notation˜Q
µ,ǫ(f) is the complexity of computing f with at most ǫ error with respect to µ, with
(rather than Q∗, as in the literature) for communication complexity in the public coin
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model. In all the above definitions, we may replace µ with U when µ is the uniform
distribution over the inputs.
The following is immediate.
Fact IV.1: For any distribution µ, number of messages c and player P,˜Qc,P
µ,ǫ(f) ≤
Qc,P
µ,ǫ(f) ≤ Qc,P
We put two constraints on protocols in the above definitions:
ǫ (f).
• We assume that the two players do not modify the qubits holding the classical input
during the protocol. This does not affect the aspect of communication we focus on in this
paper.
• We demand that the length of the i’th message sent in a protocol is known in advance.
This restriction is also implicit in Yao’s definition of quantum communication complexity
using interacting quantum circuits [35].
To illustrate this, think of a public coin classical protocol in which Alice first looks at
a public coin and if the coin is “head” sends in the first round a message of c qubits and
in the second round a message of 1 qubit, otherwise she sends one qubit in the first round
and c qubits in the second. In such a protocol the number of message bits sent in the first
round is not known in advance, and so such a protocol is not allowed in our model.
A k round protocol with communication complexity c in the more general model, in
which the restriction above is absent, can be simulated in our model losing a factor of k in
the communication complexity. To show this one invokes the principle of safe storage. The
principle says that instead of a mixed state depending on measurement results, we may
have a superposition over the measurement results and the messages. Note that in such a
superposition there may be messages of different lengths (augmented by some blanks). In
the worst case, the length of a single message is now c, so the overall communication cost
is at most kc, and the number of rounds used is always the worst case number of rounds.
In the example above we get a 2c communication complexity.
V. The Role Of Interaction In Quantum Communication
In this section, we prove that allowing more interaction between two players in a quan-
tum communication game can substantially reduce the amount of communication required.
In Section V-A we define a communication problem and formally state our results (giving
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an overview of the proof), then in Section V-B we give the details of the proofs. For
the most part, we will concentrate on communication in a constant number of rounds.
Section V-C describes the application to the disjointness problem. Section V-D discusses
our results in the case where the number of messages grows as a function of the input size.
A. The Communication Problem And Its Complexity
We define a sequence of problems S1,S2,...,Sk,... by induction. The problem S1is
the index function, i.e., Alice has an n-bit string x ∈ X1= {0,1}n, Bob has an index i ∈
Y1= [n] and the desired output is S1(x,i) = xi. Suppose we have already defined the
function Sk−1 : Xk−1× Yk−1→ {0,1}. In the problem Sk, Alice has as input her part
of n independent instances of Sk−1, i.e., x ∈ Xn
instances of Sk−1, i.e., y ∈ Yn
given to Alice if k is even and to Bob if k is odd. The output we seek is the solution to
k−1, Bob has his share of n independent
k−1, and in addition, there is an extra input a ∈ [n] which is
the a’th instance of Sk−1. In other words, Sk(x1,...,xn,a,y1,...,yn) = Sk−1(xa,ya).
Note that the size of the input to the problem Skis N = Θ(nk). If we allow k message
exchanges for solving the problem, it can be solved by exchanging Θ(logN) = Θ(klogn)
bits: for k = 1, Bob sends Alice the index i and Alice then knows the answer; for k > 1,
the player with the index a sends it to the other player and then they recursively solve
for Sk−1(xa,ya). However, we show that if we allow one less message, then no quantum
protocol can compute Sk as efficiently. In fact, no quantum protocol can compute the
function as efficiently even if we allow error, and only require small probability of error on
average.
Theorem V.1: For all constant k ≥ 1 and 0 ≤ ǫ <1
2we have
˜Qk
U,ǫ(Sk+1) = Ω?N1/(k+1)?.
To prove this theorem we prove a stronger intermediate claim. Let P1 be Bob, and
for k ≥ 2, let Pkdenote the player that holds the index a in an instance of Sk(a indicates
which of the n instances of Sk−1to solve). Let¯Pkdenote the other player. We refer to¯Pk
as the “wrong” player to start a protocol for Sk. The stronger claim is that any k message
protocol for Skin which the wrong player starts is exponentially inefficient as compared
to the logN protocol described above.
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Lemma V.2: For all constant k ≥ 1 and 0 ≤ ǫ <
Ω?N1/k?.
Indeed, there is a classical k-message, O(n)-bit protocol in which the wrong player starts,
1
2we have˜Qk,¯Pk
U,ǫ(Sk)= Ω(n)=
so our lower bound is optimal.
Theorem V.1 now follows directly.
Proof: (Of Theorem V.1): It is enough to show the lower bound for the two cases
when the protocol starts either with Pk+1or with the other player.
Let Pk+1be the player to start. Note that if we set a to a fixed value, say 1, then we
get an instance of Sk. So˜Qk,Pk+1
U,ǫ
(Sk) ≤˜Qk,Pk+1
Lemma V.2 applies.
Let player¯Pk+1be the one to start. Then, observe that if we allow one more message
(i.e., k+1 messages in all), the complexity of the problem only decreases:˜Qk+1,¯Pk+1
˜Qk,¯Pk+1
U,ǫ
(Sk+1). So we again get the bound from Lemma V.2.
U,ǫ
(Sk+1). But Pk+1=¯Pk, so the bound of
U,ǫ
(Sk+1) ≤
We prove Lemma V.2 by induction. First, we show that the index function is hard to
solve with one message if the wrong player starts. This essentially follows from the lower
bound for random access codes [13], [14]. The only difference is that we seek a lower bound
for a protocol that has low error probability on average rather than in the worst case, so
we need a refinement of the original argument. We give this in the next section.
Lemma V.3: For any 0 ≤ ǫ ≤ 1 we have˜Q1,A
Next, we show that if we can solve Skwith k messages with the wrong player starting,
U,ǫ(S1) ≥1
2(1 − H(ǫ))n.
then we can also solve Sk−1with only k − 1 messages of smaller total length, again with
the wrong player starting, at the cost of a slight increase in the average probability of
error.
Lemma V.4: For k ≥ 2 and 0 ≤ ǫ <
respect to the uniform distribution U with error ǫ, and k messages starting with¯Pk. Let
the communication complexity of P be ℓ = ℓ1+¯ℓ with ℓ1being the length of the first
message sent. Then,˜Qk−1,¯Pk−1
U,ǫ′
(Sk−1) ≤¯ℓ, where ǫ′= ǫ + 2(ℓ1/n)1/2.
We defer the proof of this lemma to a later section, but show how it implies Lemma V.2
1
2, let P be any protocol that solves Sk with
above.
Proof: (Of Lemma V.2): We prove the lemma by induction on k. The case k = 1 is
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handled by Lemma V.3. Suppose the statement holds for k−1. We prove by contradiction
that it holds for k as well. If ℓ =˜Qk,¯Pk
U,ǫ(Sk) = o(n), then by Lemma V.4 there is a k − 1
message protocol for Sk−1with the wrong player starting, with error ǫ′= ǫ + o(1) <
and with communication complexity at most ℓ = o(n). This contradicts the induction
1
2,
hypothesis.
B. The Key Lemmas
We now prove average case hardness of the index function.
Proof: (Of Lemma V.3): Consider any protocol for S1with Alice sending the first
(and only) message. Let ǫibe the probability of error when the input to Alice is uniformly
random but the input to Bob is i. Note that ǫ =?
variable containing Alice’s input, and let MB denote the qubits held by Bob after he
iǫi/n. Let X denote the random
has received Alice’s message, including his part of the shared entangled state.From
Properties (1) and (2) of mutual information in Section II-C, and the concavity of binary
entropy,
I(X : MB) ≥
?
i
I(Xi: MB) ≥
?
i
(1 − H(ǫi)) ≥ n(1 − H(ǫ)).
The second inequality follows from the fact that Bob has a measurement that predicts Xi
with error ǫiand Fact II.9 (Fano’s inequality). On the other hand, I(X : MB) is bounded
above by twice the number of qubits in the message [15, Theorem 2]. The lemma follows.
Note that for public-coin randomized protocols we do not have the factor of
1
2, and
obtain a lower bound of n(1 − H(ǫ)).
Next, we show how an efficient protocol for Skgives rise to an efficient protocol for Sk−1.
The intuition behind the argument is the same as in proofs for classical communication [10],
[36]. However, we use entirely new techniques from quantum information theory, as de-
veloped in Section III and also get better bounds.
Proof: (Of Lemma V.4): For concreteness, we assume that k is even, so that¯Pkis
Bob. Let P be a protocol that solves Skwith respect to the uniform distribution U with
error ǫ, k messages starting with Bob. Let the communication complexity of P be ℓ = ℓ1+¯ℓ
with ℓ1being the length of the first message sent.
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Given the protocol P, we devise a protocol P′for solving Sk−1 with respect to the
uniform distribution, but with Alice starting, and with only k−1 messages. The intuition
behind the protocol P′is the following. It first tries to recreate, from some shared prior
entanglement, the state after the first message in the run of P on a specially chosen Sk
instance, and then simulates the remaining k−1 rounds of communication of the protocol P
on the recreated state. The instance of Sk is such that the solution to that instance
coincides with the solution to the given Sk−1instance. We thus get a protocol for Sk−1
with the desired properties. The details follow.
We start by describing the joint pure state that Alice and Bob share in P′prior to
being given the inputs to the problem Sk−1. Consider the protocol P computing Sk. Let
MA,MB be the private qubits (or “registers”) held by Alice and Bob respectively. Let
Y = Y1Y2···Yndenote the register containing the input to Bob. Consider the state |χ? of
the registers MAMBY , after Bob sends the first message in P, when Y is initialized to a
uniform superposition over Yk= Yn
in P′is then defined as
k−1. The prior entanglement that Alice and Bob share
1
√n
n
?
j=1
|j?A|χ?AB|j?B,
where the qubits MAin |χ? are given to Alice and MB,Y to Bob. It simplifies the descrip-
tion of the protocol if Alice and Bob measure the first and the last register, respectively,
of the shared state to get a common random index j ∈ [n]. Since these registers will not
be modified during the course of the protocol, the behavior of P′is not affected by this
measurement.
We are ready to describe the steps of the protocol P′. Given the inputs x,y to Sk−1,
1. Alice, who gets the input x, initializes a register X to |φ?⊗(j−1)|x?|φ?⊗(n−j)|j?, where |φ?
is the uniform superposition over Xk−1.
Note that the state of the registers XMAMBY is now exactly as after the first message in
a run of the protocol P on an input for Skwhere a = j, all input registers Xibut for Xj
are in uniform superposition over Xk−1, Xj= x, and all Yiare in uniform superposition
over Yk−1.
2. Bob, who gets the input y, applies a unitary transformation Vj,y(to be defined below)
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to the registers MBY . This step is intended to bring the state of the registers MAMBY
close to |χ(y)?, the state after the first message in a run of the protocol P on an input
for Sk with X,Y1,Y2,...,Yj−1,Yj+1,...,Ynas above, except that register Yj is set to y
rather than the uniform superposition over Yk−1. Note that on an input as in |χ(y)?, the
result of a protocol for Skis expected to be the same as Sk−1(x,y).
3. Alice and Bob now simulate the protocol P from the second message onwards starting
with the registers XMAMBY , and declare the result of that procedure as the output of
the protocol P′.
The transformation Vj,yis defined as follows. Consider the state |χ(j,y)? of the regis-
ters MAMBY (analogous to |χ?) obtained by running P till the first message is sent, when
the register Y is initialized to |ψ?⊗(j−1)|y?|ψ?⊗(n−j), where |ψ? is the uniform superposi-
tion over Yk−1. Let ρ = TrMBY|χ??χ|, and ρj,y= TrMBY|χ(j,y)??χ(j,y)| be the restriction
of the two states to Alice. The transformation Vj,yis defined as the local unitary opera-
tor on MBY , given by Theorem II.7, that achieves the fidelity between ρ and ρj,y. This
completes the description of P′.
Observe that P′has k −1 messages starting with Alice, and has complexity¯ℓ. We now
analyze its probability of error, under a uniform distribution on inputs.
Bob’s part of the input to Skin |χ? and |χ(j,y)? differ only in the register Yj: in the
first state, this is uniform over Yk−1, whereas in the second state, this is set to y. Thus,
the state |χ? when restricted to Alice is the average encoding, over all y ∈ Yk−1, of the
state |χ(j,y)? restricted to her:
1
|Yk−1|
ρ=
?
z∈Yk−1
ρj,z.
The Average encoding theorem tells us that ρ and ρj,yare close to each other on average,
provided the mutual information µj= I(Yj: MA) between Alice’s state and Yjin a run
of P on the uniform distribution on all inputs is small:
1
|Yk−1|
?
z
h2(ρ,ρj,z)≤
?ln2
2
?
µj. (3)
As in the proof of Lemma V.3, it is not hard to see that if the length ℓ1of the first
message M is small relative to n, then for a random j, this mutual information is small.
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Claim V.5:?
By Lemma II.8, the transformation Vj,ymaps |χ? to a state close to |χ(j,y)?, and by
Lemma II.6
iµi≤ 2ℓ1. Thus, Ejµj≤ 2ℓ1/n.
?|Vj,yχ??Vj,yχ| − |χ(j,y)??χ(j,y)|?t
≤ 2√2 h(|Vj,yχ??Vj,yχ|,|χ(j,y)??χ(j,y)|)
= 2√2 h(ρ,ρj,y). (4)
For a random y ∈ Yk−1, and a random j ∈ [n], then, the average error in approximating
the state |χ(j,y)? is
Ej,y?|Vj,yχ??Vj,yχ| − |χ(j,y)??χ(j,y)|?t
≤
≤
2√2 Ej
2
√
ln2 [Ejµj]1/2
2√2 Ej,yh(ρ,ρj,y)
2√2 Ej
?Eyh2(ρ,ρj,y)?1/2
?ln2
2
From equation (4)
By Jensen’s inequality
≤
µj
?1/2
From equation (3)
≤
≤
By Jensen’s inequality
3(ℓ1/n)1/2. From Claim V.5
Running the protocol P on the input described in step 2 of P′finds Sk−1(x,y) with
probability of error at most ǫ on average when x,y are chosen at random. Thus, running
the protocol P on the state resulting from step 2 of the protocol P′gives us the answer
to Sk−1(x,y) with average probability of error only slightly higher than ǫ:
ǫ +1
2Ej,y?|Vj,yχ??Vj,yχ| − |χ(j,y)??χ(j,y)|?t
ǫ′
=≤ǫ + 2(ℓ1/n)1/2,
as claimed.
For classical randomized protocols, it is possible to simplify the reduction of Sk−1to Sk
described above: This is accomplished as follows. Recall that Alice and Bob share public
random coins. They use this to sample a (common) message m from the distribution over
classical messages in the first round of the protocol P for Sk, where the inputs are chosen
uniformly at random. They also pick a common random index j ∈ [n]. Alice now picks Xi,
i ?= j uniformly at random from Xk−1, and sets Xj= x, and a = j. Bob picks Y1,...,Yn
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from the uniform distribution over Yj−1
the protocol P on such a random input being equal to m. The distance between the joint
state so constructed and the joint state in the original protocol differs (in ℓ1-distance) by
k−1×{y}×Yn−j
k−1, conditioned on the first message in
at most the distance between Alice’s marginal distributions. Alice and Bob now simulate
the protocol P from the second message onwards on the input X,Y . A straightforward
analysis using the Average encoding theorem shows that the initial state (consisting of
the message and the inputs) constructed above differs from the corresponding state in
the protocol P by only (2ℓ1/n)1/2. This simpler argument was noted in Ref. [37] and
independently in Ref. [38].
C. The Disjointness Problem
We now investigate the bounded round complexity of the disjointness problem. Here
Alice and Bob each receive the incidence vector of a subset of a size n universe. They
reject iff the sets are disjoint. It is known [39], [12] that Q1
˜Q1
of Grover search [1]. This upper bound was later improved [20] to O(√n), although the
number of rounds remained O(√n). We now prove a lower bound by reduction.
ǫ(DISJ) ≥ (1 − H(ǫ))n and
(DISJ) = O(√nlogn) by an application
ǫ(DISJ) ≥ (1 − H(ǫ))n/2. Furthermore QO(√n)
1/3
Proof: (Of Corollary I.3): Suppose we are given a k round quantum protocol for the
disjointness problem having error 1/3 and using c qubits. W.l.o.g. we can assume Bob
starts the communication, because the problem is symmetrical, and that k is even. We
reduce the communication problem Skfrom Section V-A to DISJ.
We visualize an instance of Skas defining a subtree of the n-ary tree with k + 1 levels
and the edges at alternate levels known to Alice and Bob, respectively. The leaves of the
tree are labelled by Boolean values known to Alice (since k is even). The only edge at the
root connects it to the a’th child, where a ∈ [n] is the input that specifies which instance
of Sk−1is to be solved. The subtrees at the second level are defined recursively according
to the n instances of Sk−1.
There are at most nkpossible paths of length k that could start at the root vertex.
With each such path we associate an element in the universe for the disjointness problem.
Given the edges originating from each of their levels, Alice and Bob construct an instance
of DISJ on a universe of size N = nk. Alice checks for each possible path of length k
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whether the path is consistent with her input and whether the paths lead to a leaf which
corresponds to the bit 1. In this case she takes the corresponding element of the universe
into her subset. Bob similarly constructs his subset. Now, if the two subsets intersect,
then the (unique) element in the intersection witnesses a length k path leading to 1-leaf.
If the subsets do not intersect, then the length k path from the root leads to a 0-leaf.
We thus obtain a k round protocol for Sk in which Bob starts. By Lemma V.2, the
communication c is Ω(n) for any constant k. Since the input length for the constructed
instance of DISJ is N = nk, we get˜Qk
1/3(DISJ) = Ω(N1/k) for k = O(1).
D. Beyond A Constant Number Of Messages
So far, we have discussed the complexity of solving Skin the context of protocols with
a constant number of messages. In fact, we may derive a meaningful lower bound even
when k grows as a function of the parameter n (hence as a function of N = nk, the input
length). We may state the result as follows.
Theorem V.6: For all k = k(n) ≥ 1 and constant ǫ <
Ω?n
1
2we have˜Qk,¯Pk
U,ǫ(Sk)=
k+ k?.
Proof: Let ℓ =˜Qk,¯ Pk
U,ǫ(Sk). Then, there is a protocol that achieves this communication
complexity with ℓ1,ℓ2,...,ℓkqubits of communication in the k rounds, respectively. By
repeated application of Lemma V.4 there is a quantum protocol that solves S1with one
message, the wrong player starting, ℓkcommunication qubits and error
ǫ1
=ǫ + 2
k−1
?
?k?
?kℓ
i=1
?ℓi
n
?1/2
i<kℓi
n
?1/2
≤ǫ + 2
?1/2
By Jensen’s inequality
≤ǫ + 2
n
For a constant δ ∈ (ǫ,1
ℓ ≥ ℓk≥
we get k < 1. A contradiction. This proves that ℓ ≥ Ω(n
has at least k communication qubits and so ℓ ≥ k.
Note that this lower bound of Ω(n/k + k) also applies to classical randomized protocols.
2), if ℓ ≤ (δ−ǫ
2)2
n
kthen ǫ1 ≤ δ and by Lemma V.3 we have
2)2· 2 ·
k). Also, every k round protocol
1−H(δ)
2
n. This implies that k ≤ (δ−ǫ1
1−H(δ). For some δ close enough to ǫ
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The above theorem implies a gap in communication complexity between k and k + 1
message protocols for k up to Θ((n/logn)1/2) = Θ(logN/loglogN), and also lower bounds
for DISJ for such k.
VI. The Pointer Jumping Function
The pointer jumping function is considered in most results showing a round-hierarchy
for classical communication complexity [6], [7], [9], [8]. This problem is a particularly
natural candidate for such results.
Definition VI.1 (Pointer Jumping) Let VAand VB be disjoint sets of n vertices each.
Let FA= {fA|fA: VA→ VB}, and FB= {fB|fB: VB→ VA}, and
f(v) = ffA,fB(v) =
fA(v) if v ∈ VA,
if v ∈ VB.fB(v)
Define f(0)(v) = v and f(k)(v) = f(f(k−1)(v)).
Then gk: FA× FB→ (VA∪ VB) is defined by gk(fA,fB) = f(k+1)
fixed. The pointer jumping function fk: FA× FB→ {0,1} is the XOR of all the bits in
the output of gk.
fA,fB(v1), where v1∈ VAis
In the corresponding communication problem, Alice is given a function fA∈ FA, and Bob
a function fB∈ FB, and they are required to compute fk(fA,fB).
A. Previous Work
If Alice starts, fkhas a deterministic k round communication complexity of k logn. If
Bob starts, Nisan and Wigderson [7] proved that fkhas a randomized k round communi-
cation complexity of Ω(n
k2−k logn). The lower bound can also be improved to Ω(n
see Klauck [39]. With techniques similar to the ones in this section it is also possible to
k+ k),
show a lower bound of
(1−2ǫ)2n
2k2
−k logn for the randomized k round complexity of fkwhen
Bob starts. We omit the details.
The lower bounds are not far from the known upper bound. Nisan and Wigderson [7]
describe a randomized protocol for computing gkwith complexity O(n
klogn + klogn) in
the situation where Bob starts and k rounds are allowed. Ponzio et al. [9] show that when
k = O(1), the deterministic communication complexity of fkis O(n).
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B. A New Upper Bound
We first give a new classical upper bound which combines ideas from Nisan and Wigder-
son [7] and Ponzio et al. [9]. For n ≥ 1, define log(1)(n) = logn and for k > 1, define
log(k)(n)= log(max{log(k−1)(n),1}).
Furthermore let log∗(n) = min{k : log(k)(n) ≤ 1}.
Theorem VI.1: Rk,B
ǫ
(gk) ≤ O(klogn +
Proof: The claim is trivial for k = 1.
n
k· log1
ǫ· (log(⌈k/2⌉)(n) + logk)).
For greater k Bob starts and we have the following protocol. At the first round Bob
guesses (with public random bits) a set S0of δn random vertices from VB, we specify δ later.
For each chosen vertex v Bob communicates the first ℓ0bits of fB(v), we specify ℓ0later.
Note that the names of the chosen vertices are accessible to Alice without communication,
by reading the public random bits. The protocol then proceeds in two stages.
• Denote vt = f(t−1)(v1). For each round i = 1,...,k the active player sends vi. I.e.,
at the first round Bob sends nothing (as v1is known), at the second round Alice sends
v2= f(v1), then Bob sends f(v2) and so on. Also, at each round i Alice checks whether
vi∈ S0. Let t be the first round in which this happens. If t >k
protocol.
2the two players abort the
• The rounds t,t + 1,...,k take a special form. Let us start with round t. Alice knows
vt∈ S0and therefore knows the first ℓ0bits of fB(vt). Alice defines a set S1that contains
all elements of VAwith that prefix. I.e., |S1| ≤
Alice sends the first ℓ1bits of fA(v). In general, in the (t + i)’th round the active player
n
2ℓ0and vt+1= f(vt) ∈ S1. For each v ∈ S1
knows ℓibits of f(vt+i). The active player then defines a set Si+1that contains all the
elements of his side with that prefix. I.e., |Si+1| ≤
each v ∈ Si+1the active player sends the first ℓi+1bits of f(v).
We now specify the parameters. First we choose δ =4
n
2ℓiand vt+i+1= f(vt+i) ∈ Si+1. For
kln1
ǫ. W.l.o.g. we can assume the
vertices v2,v4,... are all distinct, or Alice can easily save two rounds and the players finish
on time. For any choice ofk
4distinct vertices v2,...,vk/2the probability, over the choice of
S0, that during the firstk
2rounds Alice will not visit S0is at most (1 −
2.
k
4n)δn≤ e−δk
4 ≤ ǫ.
So assume indeed that t ≤k
February 1, 2008 DRAFT
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27
We now chose ℓi = log(⌈k/2⌉−i)n + 3logk. It follows that for some i <
k
2we have
ℓi≥ logn and |Si| = 1 and the active player who holds vt+ialso knows f(vt+i), so he can
save two rounds and the computation ends on time.
We now count the number of communication bits. We need k logn bits for communi-
cating vi, i = 1,...,k. Also, we need?⌈k/2⌉
i=0|Si|ℓibits for communicating the first ℓibits
and so:
of each element in Si. Notice, however, that ℓi≤2ℓi−1
k2
⌈k/2⌉
?
i=0
|Si|ℓi ≤ n[δℓ0+
⌈k/2⌉
?
i=1
ℓi
2ℓi−1] ≤ n[δℓ0+1
k2
⌈k/2⌉
?
i=1
1]
= O(n
k· log1
ǫ· (log(⌈k/2⌉)n + logk))
which completes the proof.
Corollary VI.2: If k ≥ 2log∗(n) then Rk,B
1/3(gk) ≤ O((n
k+ k)logk).
C. A Lower Bound On The Quantum Communication Complexity
In this section we prove a lower bound on the quantum communication complexity of
the pointer jumping function fk, for the situation that k rounds are allowed and Bob
sends the first message. The proof uses the same ingredients as the proof of the lower
bound for the function Skin Theorem V.1, namely the Average Encoding Theorem and
the Local Transition Lemma. We will consider a quantity dtcapturing the information
the active player has in round t on vertex t + 1 of the path. This quantity will be the
informational distance between the active player’s qubits and vertex t + 1. Our goal will
be to bound dtin terms of dt−1(which is the information gain so far) plus a term related
to the average information on pointers in the other player’s input (which is low as long as
the number of qubits sent is small). This leads to a recursion imposing a lower bound on
the communication complexity, since in the end the protocol must have reasonably large
information to produce the output, and in the beginning the corresponding information d0
is 0.
Let Alice be active in the (t+1)’th round. The informational distance dt+1measures the
distance between the state of, say, Alice’s qubits together with the next vertex FB(Vt+1)
of the path, and the tensor product of the states of Alice’s qubits and FB(Vt+1). In
the product state Alice has no information about FB(Vt+1), so if the two states are close
February 1, 2008 DRAFT
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28
Alice’s powers to say something about the vertex are very limited. We will use the triangle
inequality to bound dt+1by the sum of three intermediate distances. In the first step we
move from the state given by the protocol to a state in which the (t + 1)’th vertex is
replaced by a uniformly random vertex, independent of previous communications. The
penalty we have to pay for that is proportional to dtwhich is a bound on the amount
of information Bob gained on Vt+1. We use the local transition lemma to conceal Bob’s
ability to detect such a replacement. Once the (t + 1)’th vertex is random, we deal with
the average information a player (Bob) can get on a random pointer in the other player’s
input, and this term is small when the number of communicated qubits is small. The last
step is similar to the first and reverses the first one’s effect, i.e., replaces the “randomized”
(t + 1)-th vertex by its real value again. We arrive at the desired product state.
Theorem VI.3:˜Qk,B
1/8(fk) ≥
n
2O(k)− k logn.
Note that the lower bound is linear in n for constant k and leads to Theorem I.2. It
implies a separation between the k and k + 1 round complexity of Pointer Jumping for k
upto Θ(logn) = Θ(logN), where N = nlogn is the input size.
Proof: (of Theorem VI.3) Fix a quantum protocol for fkwith probability of error1
8,
k rounds, and with Bob starting. Usually a protocol gets some classical fA and fB as
inputs, but we will investigate what happens if the protocol is started on a superposition
over all inputs, in which all inputs have the same amplitude, i.e., on
?
fA∈FA,fB∈FB
1
nn|fA?|fB?.
Note that |FA| = |FB| = nn. The superposition over all inputs is measured after the
protocol has finished, so that a uniformly random input and the result of the protocol on
that input are produced.
We also require that before round t the active player computes and measures the vertex
vt= f(t−1)(v1), and includes it in the message that is sent to the other player, who stores
it in some qubits Vt. Thus, at the first round Bob sends v0(which is known in advance)
to Alice, at the second round Alice sends v2= FA(v1) to Bob and so on. This increases
the communication by an additive k logn term. Notice that FA,FB are in a uniform
superposition over all possible inputs, and so if we don’t measure FAand FBthe register
February 1, 2008 DRAFT
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29
Viis also in a uniform superposition for every i > 1. The density matrix of the global state
of the protocol before the communication of round t is ρMA,tMB,tFAFB, where FA,FBare the
qubits holding the inputs of Alice and Bob and MA,tresp. MB,tare the other qubits in the
possession of Alice and Bob before the communication of round t. The state of the latter
two systems of qubits may be entangled. In the beginning these qubits are independent
of the input. We also denote ˜ ρMA,tMB,tFAFBthe density matrix of the system in the case
where we do not measure any of the Vi.
Let us denote dt = D2(MB,tFB : FA(Vt)) when t is odd, where the register FA(Vt)
has been measured. Notice that at this stage Vtis measured and FA(Vt) is a subregister
of FA. The quantity dt is a measure of Bob’s information on the value FA(v) Alice is
going to compute. We similarly let dt= D2(MA,tFA: FB(Vt)) when t is even, where the
register FB(Vt) has been measured.
We assume that the communication complexity of the protocol is δn and prove a lower
bound δ ≥ 2−O(k). The general strategy of the proof is induction over the rounds, to
successively bound d1,d2,...,dk+1. Bob sends the first message. As Bob has seen no
message yet, we have that I(MB,1FB: FA(V1)) = 0, and hence d1= 0. We show that
Lemma VI.4: dt+1≤ 8dt+ 4δ.
We see that dt+1≤ 9tδ for all t ≥ 0. After round k one player, say Alice, announces
the result which is supposed to be the parity of FB(Vk+1) and included in MA,k+1. On the
one hand dk+1 = D2(MA,k+1: FB(Vk+1)) ≤ 9kδ. On the other hand, by Lemma III.4(2)
D2(MA,k+1:?FB(Vk+1)) ≥ 1/8 − 1/16 = 1/16. Together,
We now turn to proving Lemma VI.4.
1
16≤ 9kδ, so δ ≥ 2−O(k).
Proof: (Of Lemma VI.4): W.l.o.g. let Alice be active in round t+1. Let MA= MA,t+1
and MB= MB,t+1. Before the t+1 round Vt+1= FA(Vt) is measured. The resulting state is
a probabilistic ensemble over the possibilities to fix V1,...,Vt+1, which are then classically
distributed. Alice’s reduced state is block diagonal with respect to the possible values of
the vertices V1,...,Vt+1. For any value v of Vt+1let ρv
MAMBFAFB= ρVt+1=v
MAMBFAFBdenote the
pure state with vertex Vt+1fixed to v. Our first goal is to bound the amount of information
Bob has at this stage about Alice’s value Vt+1. We define:
February 1, 2008DRAFT
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30
γv
def
=h2?ρv
MBFB,ρMBFB
?.
I.e., we look at Bob’s view before the t+1 message, and in particular before Alice sends
Vt+1to him, and we let γvmeasure how much Bob’s view when Vt+1= v differs from Bob’s
average view. We show that these two are typically close to each other, namely:
Lemma VI.5: Evγv≤ dt.
Loosely speaking this says that Bob does not know more than dt units of information
about FA.
The next step is to replace the actual state ρv
MAMBFAFBwhere Vt+1= v with the average
case ρMAMBFAFBRwhere nothing is known about Vt+1. As we saw, typically, Bob can not
distinguish between the actual encoding and the average one, so this should not matter
much to Bob. We let ρv
MAMBFAFBRbe a purification of ρv
MAMBFAFBwhere R is some addi-
tional space used to purify the random path V1,...,Vt. I.e., ρv
MAMBFAFBRreflects a purifi-
cation of Bob’s view, when Vt+1= v. We let ρMAMBFAFBRbe a purification of ρMAMBFAFB
where R is some additional space used to purify the random path V1,...,Vt+1. Now,
due to Lemma II.8 there is a local unitary transformation Uv acting only on FAMAR
such that σv
MAMBFAFBR
def
=UvρMAMBFAFBRU†
v, and ρv
MAMBFAFBRare close to each other.
σv
MAMBFAFBRreflects a purification of Bob’s average view with Alice locally adding Vt+1= v
to it . Notice that in σv
MAMBFAFBR, v is arbitrary and in particular can be different than
Vt+1. By Lemma II.8 for all vertices v ∈ VB,
h2?ρv
MAFA,σv
MAFA
?
≤ h2?ρv
≤ h2?ρv
= h2?ρv
MAFAFB(v),σv
MAFAFB(v)
?
MAMBFAFBR,σv
MAMBFAFBR
?
MBFB,ρMBFB
?= γv, (5)
We are interested in the value
dt+1
=D2(MAFA: FB(Vt+1)) = Ev h2?ρv
where FB(v) is measured and the expectation is over the uniform distribution on ver-
MAFAFB(v),ρv
MAFA⊗ ρFB(v)
?,
tices v. We now study this expression under the average case scenario, i.e., we look at
h2?
σv
MAFAFB(v),σv
MAFA⊗ ρFB(v)
?
. We prove:
February 1, 2008DRAFT
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31
Lemma VI.6: For all vertices v ∈ VB,
h2?σv
MAFAFB(v),σv
MAFA⊗ ρFB(v)
?
≤
˜dt+1(v)
where,
˜dt+1(v)
def
=h2?˜ ρMAFAFB(v), ˜ ρMAFA⊗ ρFB(v)
?,(6)
where FB(v) is assumed to have been measured. Recall that in ˜ ρ we let V1,...,Vt+1go
unmeasured and that v is an arbitrary value not necessarily equal to Vt+1. We then prove:
Lemma VI.7: Ev˜dt+1(v) ≤ 2δ.
Assuming the above lemma, we see that for all v:
h?ρv
≤
MAFAFB(v),ρv
MAFA⊗ ρFB(v)
?
h?ρv
+ h?σv
+ h?σv
MAFAFB(v),σv
MAFAFB(v)
?
MAFAFB(v),σv
MAFA⊗ ρFB(v)
MAFA⊗ ρFB(v)
MAFAFB(v),σv
?
MAFA⊗ ρFB(v),ρv
2√γv+ h?σv
2√γv+
˜dt+1(v)
?
≤
MAFA⊗ ρFB(v)
?
From equation (5)
≤
?
From Lemma (VI.6).
Squaring both sides,
h2?ρv
MAFAFB(v),ρv
MAFA⊗ ρFB(v)
?
≤
?
8γv+ 2˜dt+1(v).
2√γv+
?
˜dt+1(v)
?2
≤ (7)
I.e., we paid an 8γvpenalty, and we switched to the scenario where Bob has no information
about Vt+1. Now,
D2(MAFA: FB(Vt+1))=
Ev h2?ρv
Ev[8γv+ 2˜dt+1(v)]
MAFAFB(v),ρv
MAFA⊗ ρFB(v)
?
≤
≤
By equation (7)
8dt+ 4δ By Lemma VI.7.
This completes the proof of Lemma VI.4.
February 1, 2008DRAFT
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32
We finish the proof of Theorem VI.3 by proving the remaining Lemmas.
Proof: (Of Lemma VI.5): By definition Evγvis Euh2?
D2(MBFB: FA(Vt)). Now, D2(MB,t+1FB: FA(Vt)) ≤ D2(MB,tFB: FA(Vt)) = dtbecause
Bob sends the t’th message, and this only decreases the informational distance.
ρVt=u
MBFBFA(u),ρVt=u
MBFB⊗ ρFA(u)
?
=
Proof: (Of Lemma VI.6):
h2?σv
≤
=
MAFAFB(v),σv
MAFA⊗ ρFB(v)
MAFAR⊗ ρFB(v)
?
h2?σv
h2?ρMAFARFB(v),ρMAFAR⊗ ρFB(v)
h2?˜ ρMAFAFB(v), ˜ ρMAFA⊗ ρFB(v)
˜dt+1(v)
MAFARFB(v),σv
?
?
By unitarity
=
?
(8)
= By definition (6).
For equation (8), notice that R holds the path V1,...,Vt+1, which is determined by MAFA.
We can apply a unitary transformation that “erases” this. We then get a pure state that
is ρ with V1,...,Vt+1unmeasured, i.e., what we called ˜ ρ
Proof: (Of Lemma VI.7): We first bound the information Alice has on Bob’s input.
For all t, I(MA,tFA: FB) is bounded above by twice the number of qubits in the messages
so far due to Lemma II.10, assuming that FB is measured, i.e., I(MA,tFA: FB) ≤ 2δn.
Thus considering the situation that FB is distributed uniformly instead of being in the
uniform superposition we get EvI(MAFA: FB(v)) ≤ 2δ (where v is uniformly random),
using Equation (1) and that the FB(v) are mutually independent. Now,
Ev˜dt+1(v) = Evh2?˜ ρMAFAFB(v), ˜ ρMAFA⊗ ρFB(v)
= EvD2(MAFA: FB(v)),
?
where FAMAMBFBare as in the protocol without measurements. Also I(MAFA: FB(v))
is invariant if FB(i) is in superposition or measured for i ?= v. So,
EvD2(MAFA: FB(v))≤
=
EvI(MAFA: FB(v)) By Lemma III.3
2δ.
February 1, 2008DRAFT
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33
Acknowledgements
We thank Jaikumar Radhakrishnan and Venkatesh Srinivasan for their input on the
classical communication complexity of Pointer Jumping and the subproblem Sk, Dorit
Aharonov and Pranab Sen for helpful feedback on earlier versions of the paper, and Elitza
Maneva and Leonard Schulman for discussions on applying our techniques to classical
protocols for Sk. We thank the anonymous referee for useful comments.
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