# Interaction in Quantum Communication and the Complexity of Set Disjointness

**ABSTRACT** One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much 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. We consider the question as to whether every classical protocol may be transformed to a simpler" quantum protocol|one that has similar eciency, but uses fewer message exchanges.

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**ABSTRACT:**Quantum Shannon theory is loosely defined as a collection of coding theorems, such as classical and quantum source compression, noisy channel coding theorems, entanglement distillation, etc., which characterize asymptotic properties of quantum and classical channels and states. In this paper, we advocate a unified approach to an important class of problems in quantum Shannon theory, consisting of those that are bipartite, unidirectional, and memoryless.IEEE Transactions on Information Theory 11/2008; · 2.62 Impact Factor - SourceAvailable from: psu.eduElectronic Colloquium on Computational Complexity (ECCC). 01/2011; 18:62.
- SourceAvailable from: uni-frankfurt.de[Show abstract] [Hide abstract]

**ABSTRACT:**This is an excerpt from my paper "On quantum and approximate privacy", published in Theory of Computing Systems vol.37(1), pp.221-246, 2004 (previous version in STACS 2002), and criticized by Jakoby et al. at this Dagstuhl workshop. Note that for the purpose of refuting Jacoby et al.'s claim that all functions can be computed privately in the quantum case it would suffice to consider the matrix for the two-bit Boolean AND in the proof of Theorem 2, for a more convenient argument. Moreover we sketch at the end how Jakoby et al.'s oblivious trans-fer protocol (starting on page 18 of the slides made available on this site, see the talk by Maciej Liskiewicz) fails against a simple EPR at-tack. Essentially their mistake is to ignore Alice's ability to keep a purification of her "random bits" instead of simply tossing coins, an attack that is undetectable for Bob and hence allowed in our definition of privacy.

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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)

February 1, 2008 DRAFT

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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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