Experimental multipartner quantum communication complexity ...
Nat Comput (2013) 12:19–26 DOI 10.1007/s11047-012-9352-7
Experimental multipartner quantum communication complexity employing just one qubit Pavel Trojek • Christian Schmid • Mohamed Bourennane ˇ aslav Brukner • Marek Zukowski _ • C Harald Weinfurter
•
Published online: 5 January 2013 Ó Springer Science+Business Media Dordrecht 2013
Abstract Most proposals for quantum solutions of information-theoretic problems rely on the usage of multi-partite entangled states which are still difficult to produce experimentally with current state-of-the-art technology. Here, we analyze a scheme to simplify a particular kind of multiparty communication protocols for the experiment. We prove that the fidelity of two communication complexity protocols, allowing for an N - 1 bit communication, can be exponentially improved by N - 1 (unentangled) qubit communication. Taking into account, for a fair comparison, all inefficiencies of state-of-the-art set-up, the experimental implementation for N = 5 outperforms the best classical protocol, making it the candidate for multi-party quantum communication applications. Keywords Communication complexity Quantum communication Quantum information Parametric down conversion P. Trojek (&) C. Schmid H. Weinfurter Ludwig-Maximilians-Universita¨t, 80799 Mu¨nchen, Germany e-mail: [email protected] P. Trojek C. Schmid H. Weinfurter Max-Planck-Insititut fu¨r Quantenoptik, 85748 Garching, Germany M. Bourennane Physics Department, Stockholm University, 10691 Stockholm, Sweden Cˇ. Brukner Institut fu¨r Experimentalphysik, Universita¨t Wien, Boltzmanngasse 5, 1090 Wien, Austria _ M. Zukowski Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gdanski, 80-952 Gdansk, Poland
1 Introduction Quantum information science overcomes a number of barriers for conventional information transfer, cryptography and computation. However, so far the only quantum scheme that reached the stage of commercial application is quantum key distribution (Bennett and Brassard 1984). All other quantum information processes, to reach this stage, require the simplification of the quantum-assisted protocols and/or development of new experimental techniques for controlling and processing of quantum information. In particular multi-party quantum communication protocols require multipartite entangled states, which are very difficult to be produced with current methods and moreover suffer from a high noise. In the domain of photonic qubits, the yield of multi-partite entangled states from, currently used, spontaneous parametric down conversion (SPDC)schemes decreases exponentially with the number of partners. It was shown recently that entanglement is not the only non-classical resource endowing the quantum multiparty information processing its power. Instead, even the sequential communication and transformation of a single qubit can be sufficient (Galva˜o 2002; Buhrman et al. 1998; Raz 1999). This breakthrough makes multi-party communication tasks feasible, and remarkably, technologically comparable to quantum key distribution as we are going to show in this contribution. One of the first significant advantages of introducing quantum phenomena to communication problems was recognized in the field of communication complexity problems (CCP’s) (Yao 1979). There the goal is to find the minimum amount of information, which parties, performing some local computations, have to exchange in order to accomplish some globally defined goal. For example, in a two-party CCP, the parties P 1 and P 2 receive each an n-bit
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string X1 and X2 with the common goal to compute the value of a given function T(X1,X2). Before distribution of the strings, the parties are allowed to communicate freely and prepare themselves jointly in any way, e.g., by sharing classically correlated random variables, agreeing on a common strategy etc. However, once received, they must communicate as little as possible. The question is how many bits of information exchange then is enough to compute T(X1,X2) (Buhrman et al. 1998, 1999; Cleve et al. 1999), or alternatively how high is the probability to reach the correct value of T(X1,X2) under a specific communication restriction (Buhrman et al. 2001; Brukner et al. 2002, 2004; Galva˜o 2002). The purpose of such studies is making distributed computations efficient in terms of communication, and thus speeding them up. Besides, such an abstract problem finds its practical use in many other applications, including the design of Very Large Scale Integrated circuits or study of data structures (Kushilevitz and Nisan 1997). Quantum protocols involving multiparty entangled states were shown to be superior to classical protocols for a number of CCPs (Buhrman et al. 2001; Cleve et al. 1999; Buhrman et al. 1999; Hardy and van Dam 1999). As a criterium of the advantage, the violation of a Bell inequality, related with the task of the protocol, can be used (Brukner et al. 2002, 2004). Despite many theoretical results, with exception of the experiment that we report in the paper, no experiment showing a genuine quantum advantage in solving CCPs has been reported; see also feasibility studies (Galva˜o 2002; Cabello and Lo´pez-Tarrida 2005). The reason is the above mentioned problem concerning the generation of multi-partite entangled states. The only laboratory demonstrations so far, corrected the measured results for the experimental inefficiencies (Xue et al. 2001; Horn et al. 2005; Zhang et al. 2007), and thus did not provide the conclusive and unambiguous practical manifestation of the superiority of quantum-assisted protocols in distributed computations. Here we show that, for solving CCPs with restricted communication, the usage of entangled states is not required and that single qubit assisted protocols are indeed superior over the corresponding classical ones. Their superiority may increase even exponentially with the number of partners. We review the experimental results obtained in (Trojek et al. 2005), where we could demonstrate by using parametric down-conversion as a source of heralded single qubits that quantum protocols solve two CCPs more efficiently, even with the limited detection efficiency inherent in any real experiments. We focus in the present work on the underlying theory and present the theoretical study which includes the explicit derivation of the upper classical bounds on success probability for the two examples of CCPs. Thereby we provide a rigorous proof of the exponential separation of
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quantum/classical solutions in terms of their efficiency. Finally we point out that our single qubit approach is generic for the solution of any communication protocol relying on the correlation function of GHZ states.
2 Theory 2.1 Communication complexity problems Let us introduce the two CCPs which we will analyze here and later on show their experimental implementations. The first one, problem A, is the so-called modulo-4 sum problem. It was first defined for three parties by Buhrman et al. (2001) and later generalized to N parties in (Buhrman et al. 1999). The problem in the setting with restricted communication is stated as follows: Imagine N separated partners P 1 ; . . .; P N . Each of them receives a two-bit input string Xk ; ðXk ¼ 0; 1; 2; 3; k ¼ 1; . . .; NÞ: The Xks are distributed such that their sum is even, P i.e., ( Nk=1 Xk)mod 2 = 0. No partner has any information whatsoever on the values received by the others. Next, the partners communicate with the goal that one of them, say P N ; can tell whether the sum modulo-4 of all inputs is equal zero or two. That is, P N should announce the value of a dichotomic, i.e. Pof values ±1, function TðX1 ; . . .; XN Þ given by TA = 1 ( Nk=1 Xkmod 4). The total amount of communication is restricted to only N - 1 bits (classical scenario). The partners can freely choose a communication protocol as long as it does not depend on input data. Such a dependence would imply a violation of the communication restriction. (E.g. they can choose between sequential communication from one to the other, or any arbitrary tree-like structure ending at the last party P N :) An alternative description of the task A, simplifying the calculation of the maximum classical success probability and making the connection with the task B more visible, can be introduced. It puts the probability distribution for local data as ! N 1 pX pA ðX1 ; . . .; XN Þ ¼ 2N1 cos ð1Þ Xk ; 2 2 k¼1 and the global task function as ! N pX TA ðX1 ; . . .; XN Þ ¼ cos Xk : 2 k¼1
ð2Þ
Problem B has a similar structure as A, but now N real numbers X1 ; . . .; XN 2 ½0; 2pÞ with probability density pB ðX1 ; . . .; XN Þ ¼
1 4ð2pÞN1
jcosðX1 þ þ XN Þj
ð3Þ
are distributed to the partners. Their task is to compute whether cosðX1 þ . . . þ XN Þ is positive or negative, i.e. to give the value of the dichotomic function
Experimental multipartner quantum communication complexity
"
N X
TB ¼ S cos
21
!# Xk
;
ð4Þ
k¼1
where S(x) = x/|x|. The communication restriction is the same as for problem A, i.e., only N - 1 bits are allowed to be exchanged. 2.2 Optimal classical protocol To find the best performing protocols for these two CCPs, it is convenient to first rewrite the random inputs Xks. For the task A we put Xk = (1 - yk) ? xk, where yk 2 f1; 1g; xk 2 f0; 1g: For the task B we write Xk = p (1 yk)/2 ? xk, with yk 2 f1; 1g; xk 2 ½0; pÞ: Accordingly, the task function T can now be reformulated in a common Q form: T ¼ f ðx1 ; . . .; xN Þ Nk¼1 yk and pðX1 ; . . .; XN Þ ¼ 2N p0 ðx1 ; . . .; xN Þ: This reformulation enables one to adopt the mathematical formalism developed for entanglement-assisted communication complexity models in (Brukner et al. 2002, 2004). Specifically, for the task A P we have f ¼ fA ¼ cosðp2 Nk¼1 xk Þ with p0 ¼ p0A ¼ 2Nþ1 P j cos p2 Nk¼1 xk j; and for the task B; f ¼ fB ¼ P P S½cosð Nk¼1 xk Þ with p0 ¼ p0B ¼ 21 pNþ1 j cosð Nk¼1 xk Þj. We note that the dichotomic variables yk are not restricted by the probability distributions, p, for the Xks. Thus, they are completely random. Furthermore, since T is proportional to the product of all yks, the answer eN = ± 1 of P N is completely random with respect to T, if it does not depend on every yk. Thus, an unbroken communication structure is necessary: the information from all N - 1 partners must directly or indirectly reach P N : Due to the restriction to N 1 bits of communication each of the partners, P k ; where k ¼ 1; . . .; N 1; sends only a one-bit message, which for convenience will be denoted as ek = ±1. The task function T as well as the answer eN of P N takes only two values, ±1. If the answer is correct, then T = eN and thus TeN = 1; otherwise, TeN = -1. Therefore, following (Brukner et al. 2004] we can quantify the average P success of the protocol with the fidelity F ¼ X1 ;...;XN pTeN ; or equivalently F¼
X 1 p0 ðx1 ; . . .; xN Þf ðx1 ; . . .; xN Þ N 2 x ;...;x ¼0;1 1
N
X
N Y
ð5Þ
yk eN ðx1 ; . . .; xN ; y1 ; . . .; yN Þ
directly from partners P i1 ; . . .; P i‘ : That is, eN ¼ eðxN ; yN ; ei1 ; . . .; ei‘ Þ: Let us fix xN, and treat e as a function ex_N of the remaining ‘ ? 1 dichotomic variables, yN ; ei1 ; . . .; ei‘ : Such functions can be represented as 2‘?1 dimensional vectors. These, in turn, can be expanded with the use of an orthogonal basis given by Vjj1 ...j‘ Q ðyN ; ei1 ; . . .; ei‘ Þ ¼ yjN ‘k¼1 ejikk ; where j; j1 ; . . .; j‘ ¼ 0; 1: Therefore, one can express ex_N in the following way: cjj1 ...j‘ ðxN ÞyjN
j;j1 ;...;j‘ ¼0;1
‘ Y
ejikk ;
ð6Þ
k¼1
where cjj1 ...j‘ ðxN Þ ¼
1 2‘þ1 y
X
exN V jj1 ...j‘ :
ð7Þ
N ;ei1 ;...;ei‘ ¼1
Since jexN j ¼ jVjj1 ...j‘ j ¼ 1; one has jcjj1 ...j‘ ðxN Þj 1: We put P 0 the expansion to Eq. (5). As, y_N= ± 1yNyN = 0, and P 0 y_k = ± 1ykek = 0, only the term with j; j1 ; . . .; j‘ ¼ 1 in expansion (6) can give a non-zero contribution to Fc. Thus, without changing the result of Eq. (5), eN in (5) can Q be replaced by a function e0N ¼ yN cN ðxN Þ ‘k¼1 eik : where cN(xN) stands for c11...1 ðxN Þ: Next, notice that, e.g., ei_1, which is in the formula for e0 N, can depend only on xi_1, yi_1 and the messages obtained by P i1 from a subset of partners: ep1 ; . . .; epm (this set does not contain any ei_k). In analogy with (6), ei_1, for a fixed xi_1, can be expanded in terms of orthogonal basis functions: ei1 ¼
X
c0jj1 ...jm ðxi1 Þyji1
j;j1 ;...;jm ¼0;1
m Y
ejpkk :
ð8Þ
k¼1
Again, jc0jj1 ...jm ðxi1 Þj 1: If one puts this into e0 N one obtains for the fidelity 1 X Fc ¼ N2 gðx1 ; . . .; xN ÞcN ðxN Þci1 ðxi1 Þ 2 x1 ;...;xN ð9Þ m ‘ X Y Y Y yk epr eik ; y0 k6¼N;i1
r¼1
k¼2
P where g ¼ p0 f ; ci1 ðxi1 Þ ¼ c011...1 ðxi1 Þ; and y’ represents summation over y1 ; . . .; yi1 1 ; yi1 þ1 ; . . .; yN1 ¼ 1: Note that each message appears in the product only once. We continue this procedure of expanding the messages, till it halts (i.e., till we reach the level of those partners who do not receive any messages). The end result is
y1 ;...;yN ¼1 k¼1
The probability of success reads P = (1 ? F)/2. For the problem B integrations replace summations Rp P ð xk ! 0 dxk Þ: In any classical protocol the answer eN given by P N can depend on yN, xN, and on the messages, ei1 ; . . .; ei‘ ; received
X
e xN ¼
Fc ¼
X x1 ;...;xN
gðx1 ; . . .; xN Þ
N Y
cn ðxn Þ;
ð10Þ
n¼1
with |cn(xn)| B 1. Since Fc in Eq. (10) depends on the product of local functions cn(xn), it has to be bounded from above, i.e., jFc j BðNÞ; where BðNÞ is the maximum
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classical fidelity. The reasoning is similar to the one leading to Bell inequalities; (as a matter of fact the inequality jFc j BðNÞ; where Fc is defined in (10), is an algebraic version of a general Bell inequality for N observers (Brukner et al. 2002, 2004]). The extrema of Fc are at the limiting values cn(xn) = ± 1, because Fc is linear in every cn(xn). It becomes apparent now that the class of protocols, in which the partners P 1 to P N1 calculate ek = ykck(xk), with ck(xk) = ± 1, and send the Q result, encoded in ek, to P N ; who puts eN ¼ Nk¼1 yk ck ðxk Þ; contains an optimal protocol of fidelity BðNÞ: Let us calculate the specific forms of classical fidelity bounds BðNÞ for both problems. With the use of formulas (1) and (2) we obtain for the problem A the following form of Fc,A:
Fc;B ¼
1 2pN1
Zp
Zp dx1 . . .
dxN cosðx1 þ . . .
0" 0 !# N N Y X xk ck ðxk Þ: þ xN Þ S cos k¼1
ð17Þ
k¼1
Since jyj S½y ¼ y and |ck(xk)| = 1, one can derive the _ following inequality (Zukowski 1993): ! p p Z Z N N Y X dx1 . . . dxN cos xk ck ðxk Þ 2N : ð18Þ 0
0
k¼1
k¼1
ð12Þ
The derivation goes as follows. One expands the cosine function, and notices that the intergrations over xi are always Rp of the following form 0 dxi bðxi Þci ðxi Þ; where b(xi) is sin xi or Rp cos xi : It is easy to notice that if 0 dxi ci ðxi Þ sin xi ¼ Bi sin ai Rp then one must have 0 dxi ci ðxi Þ cos xi ¼ Bi cos ai ; where ai Rp is a certain angle, and Bi ¼ maxb 0 dxi sin ðxi þ bÞci ðxi Þ: Of course Bi B 2. Putting all that into the left hand side of the inequality (18), we get straightforwardly the right hand side bound. Therefore, the classical bound BðNÞ for the task B involving any number of parties N is given by N1 1 2 N BðNÞ ¼ maxðFc;B Þ ¼ N1 2 ¼ : ð19Þ 2p p
Since ck are dichotomic functions of spectrum ±1, the possible values of (12) are p pffiffiffi p 1 i ¼ 2 exp i þ in ; ð13Þ 4 2
In both cases the fidelity decreases exponentially with the number of parties N. For the task A we found Fc,A B 2-K?1, where K = N/2 and K = (N ? 1)/2 for even and odd numbers of parties, respectively. This analytic result confirms the numerical simulations of (Galva˜o 2002) for small N. For the task B we derived Fc,B B (2/p)N-1.
Fc;A
! N N Y pX ¼ N1 cos xk ck ðxk Þ 2 2 k¼1 x1 ;...;xN k¼1 " # N X p Y 1 ¼ N1 Re exp ixk ck ðxk Þ : 2 2 k¼1 xk 1
X
ð11Þ
Each of the numbers x1 ; . . .; xN takes only the values zero and one, so the sum in (11) reduces to X xk
p exp ixk ck ðxk Þ ¼ ck ð0Þ þ ick ð1Þ: 2
where n is an integer. Considering an even number N of parties, i.e. N = 2 K, K being an integer C1, the expression in (11) is maximized by using pairs of conjugate values (13). Thus the maximum is given by BðN ¼ 2KÞ ¼ maxðFc;A Þ " # K pffiffiffi p p Y 1 ppffiffiffi p 2 exp i in 2 exp i þ in ¼ 2K1 Re 2 4 2 4 2 i¼1 ð14Þ and one can easily show that the bound for even number of parties is BðN ¼ 2KÞ ¼ 2Kþ1 :
ð15Þ
Analogously, for odd number of parties, i.e. N = 2K - 1, we can find the bound BðN ¼ 2K 1Þ ¼ 2Kþ1 : For the problem B, the fidelity Fc,B is given by
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ð16Þ
2.3 Quantum protocol For the quantum protocols, we note that the Holevo (1973) bound limits the information storage capacity of a qubit to no more than one classical bit. Thus, we must now restrict the communication to N - 1 qubits, or alternatively, to N - 1-fold exchange of a single qubit. The optimal solution of the task A starts with a qubit pffiffiffi in the state j w0 i ¼ 1= 2ðj 0iþ j 1iÞ: Parties then sequentially act on the qubit with the phase-shift transformation j 0ih0 j þeipXk =2 j 1ih1 j; in accordance with their local data Xk. After all N phase shifts the state takes the form: PN 1 j wN i ¼ pffiffiffi j 0i þ eipð k¼1 Xk Þ=2 j 1i : ð20Þ 2 Since the sum over Xk is even, the phase factor PN eipð k¼1 Xk Þ=2 is equal to the dichotomic function TA to be computed. Thus, a measurement of the qubit in the basis
Experimental multipartner quantum communication complexity
23
pffiffiffi ðj 0i j 1iÞ= 2 reveals the value of TA with fidelity Fq,A = 1, that is, always correctly. Task B starts also with a qubit in the state j w0 i: Each party performs according to his/her local data a unitary transformation j 0ih0 j þeiXk j 1ih1 j; leading to PN 1 ð21Þ j wN i ¼ pffiffiffiðj 0i þ ei k¼1 Xk j 1iÞ: 2 The last party makes the same measurement as in the task pffiffiffi A. The probability for the detection of state 1= 2ðj 0i j 1iÞ; which we associate with the result r = ±1, is given P by PðÞ ¼ ½1 cosð Nk¼1 Xk Þ=2: The expectation value for the final answer eN = r is E = P(? ) - P(-), and P reads cosð Nk¼1 Xk Þ: The fidelity of eN, with respect to TB is Fq;B ¼
Z2p
Z2p dX1 . . .
0
dXN pB ðX1 ; . . .; XN ÞTB ðX1 ; . . .; XN Þ
0
EðX1 ; . . .; XN Þ:
ð22Þ
With the actual forms of pB, TB, and E, one gets Fq,B = p/ 4, i.e., the protocol gives the correct value of TB with probability Pq,B = (1 ? p/4)/2 & 0.892. Thus, the simple, qubit assisted quantum protocols are proven to be significantly more efficient than any classical protocols for both problems. Expressing the efficiency of protocols in terms of fidelities even an exponential separation for classical/quantum scenario is obtained, i.e. the ratio Fc/Fq decreases exponentially with N! Moreover, Fq for both problems are independent of N.
3 Experiment We implemented the quantum protocols for N = 5 parties (Trojek et al. 2005), using a heralded single photon as the carrier of the qubit communicated sequentially by the partners. The qubit was encoded in the polarization, so that the computational basis, ‘‘0‘‘ and ‘‘1’’, corresponds to horizontal H and vertical V linear polarization, respectively. The data Xk of each party was encoded on the qubit via a phase shift, using birefringent materials. The last pffiffiffi party performed a measurement in the basis 1= 2ðj Hi j ViÞ to obtain the answer eN. The experimental set-up is shown in Fig. 1. Time-correlated photon pairs are produced via non-collinear SPDC process in a 2-mm-long BBO crystal pumped by a blue LD (kp = 402.5 nm) with the output optical power of about 10 mW. The detection of one photon by the trigger detector DT heralds the existence of the other one used in the protocol. The narrow gate window of 4 ns for coincidence detection between these two photons, along with the singlecount rates of & 140,000 s-1 at the detectors D? and D-,
warrant that the recorded data are due to single photons; the probability of having more than one photon per heralding signal and during time corresponding to the gate window is estimated to 2 9 10-3, assuming the given parameters and the Poissonian photon-number statistics of SPDC photons. Since a type II degenerate phase-matching scheme is used, the emitted photon pairs at k = 805 nm (Dk 6 nm) are orthogonally polarized. Filtering of the vertical polarization for the trigger photons, ensures that the protocol photon has horizontal polarization initially. A half-wave plate (HWP1) transforms the qubit to the initial state pffiffiffi 1= 2ðj Hiþ j ViÞ as required by the optimal quantum protocol. The individual phase shifts of parties are implemented using a 200 lm thick Yttrium-Vanadate (YVO4) birefringent uniaxial crystals (Ci). The crystals, cut with their optic axes parallel to the surface, are aligned in such a way that H and V polarization states correspond to their normal modes. In this configuration the length difference between optical paths of horizontal and vertical polarization can be continuously tuned using motor-driven rotations of the crystals along their optic axes, thereby allowing to set any desired phase-shift from 0 to 2p independently from the incoming polarization state. The precision in applying an arbitrary phase shift was measured to be better than 0.02p over the full phase range. An additional YVO4 crystal (Ccomp, 1,000 lm long), aligned with its optic axis in the plane perpendicular to the direction defined by the optic axes of the previous crystals, is used to compensate dispersion effects. To analyze the polarization state of photons pffiffiffi in the basis ðj 0i j 1iÞ= 2; a half wave-plate (HWP2) at an angle of 22.5° followed by polarizing beam-splitter (PBS) is used. For a fair comparison between the quantum protocols the classical one-bit protocols, no heralded events are
RNG
RNG
RNG
RNG
RNG
Fig. 1 Experimental set-up for solving of qubit-assisted CCPs. Pairs of orthogonally polarized photons are emitted from a BBO crystal via the type-II SPDC process. The detection of one photon by the trigger detector DT indicates the existence of the protocol photon. The polarization state is prepared with a half-wave plate (HWP1) and a polarizer, placed in the trigger arm. According to the local data (X1 to X5), each of the parties introduces a phase-shift by the rotation of a birefringent YVO4 crystal (C1–C5). The last party performs the analysis of a photon-polarization state using a half-wave plate (HWP2) followed by a polarizing beam-splitter (PBS)
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discarded, even if the detection of the protocol photon fails. In such a case one can still guess the value of T, but with success rate of only 1/2. Therefore, a high detection efficiency g of the heralded photons, i.e. high coincidence/ single ratio for our set-up, is essential for an unambiguous demonstration of the superiority of the qubit-assisted protocol. E.g., in a realistic implementation of the five-party problem A, the efficiency of gJ0:33 was estimated to be sufficient to beat the optimum classical protocol, see (Galva˜o 2002). To minimize the cases with no detection of the photon, the yield of heralded photons was maximized by adopting an unbalanced SPDC scheme. We select a restricted spatial mode with well defined polarization of the trigger photons by coupling them into a single-mode fiber behind a polarizer, whereas no spatial filtering is performed on the protocol photons. As a result, we observed & 5000 trigger events per second at the detector DT with & 2,400 coincident events per second of protocol detections, i.e. an overall detection efficiency of g& 0.48, close to the limit given by the detector efficiency of our silicon APDs (measured to be about & 55 %). The protocols were run many times, to obtain sufficient statistics. Each run took about one second. It consisted of generating a set of pseudorandom numbers obeying the specific distribution, subsequent setting of the corresponding phase shifts, and opening detectors for a collection time window s. The limitation of communicating one qubit per run requires that only these runs, in which exactly one trigger photon is detected during s, are selected for the evaluation of the probability of success Pexp. To maximize the number of such runs, n, the length of s was optimized to 200 ls, assuming a Poissonian photon-number distribution of SPDC photons. We merely remark that essentially the same set-up could be used to solve the CCPs using bright polarized pulses. However, in such a case, a suitable polarization measurement of the pulses reveals all the encoded input data of any party: two bits for the task A, and arbitrarily many for the task B. Thus, the communication restriction to N - 1 bits is violated. Attenuation of the pulses to the single-photon level does not help either. The efficiency of the protocol is significantly lowered in such a case, due to many nondetection events, forcing one to guess the answer most of the time.
4 Results In order to determine the probability of success from the data acquired during the runs we have to distinguish the following two cases. First, the heralded photon is detected, which happens with probability g, given by the
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coincidence/single ratio. Then, the answer eN can be based on the measurement result. However, the answer is correct only with a probability c, due to experimental imperfections in the preparation of the initial state, the setting of the desired phase shifts, and the polarization analysis. This must be compared with the theoretical limits given by Pq,A and Pq,B for the task A and B, respectively. Second, with the probability 1 - g the detection of the heralded photon fails. Forced to make a random guess, the answer is correct in half of the cases. This leads to an overall success probability Pexp = gc ? (1 - g)0.5, or a fidelity of Fexp = g(2c - 1). Due to a finite measurement sample, our experimental results for the success probability are distributed around the value Pexp as shown in Fig. 2 for both tasks. [tb] The width of the distribution is interpreted as the error in the experimental success probability. For the task A we obtain a quantum success probability of Pexp,A = 0.711 ± 0.005. The bound Pc,A = 5/8 of the optimal classical protocol for N = 5 parties is violated by 17 standard deviations. For the task B we reached Pexp,B = 0.669 ± 0.003, whereas the classical bound is Pc,B & 0.582. The violation is by 29 standard deviations. Expressing the final results in terms of fidelities, we obtain Fexp,A = 0.421 ± 0.010 for the task A, and Fexp,B = 0.337 ± 0.006 for the task B. This must be compared to the performance of the best classical protocol for N = 5 parties reaching Fc,A = 0.25 for A and Fc,B&0.164 for B. Table 1 summarizes the relevant experimental parameters for both tasks.
(a)
(b)
Fig. 2 Histograms of measured quantum success probabilities a for the task A and b for task B. The bounds for optimum classical protocols are displayed, too
Table 1 Relevant experimental parameters in the implementation of communication complexity problems A and B: n number of correct runs, g overall detection efficiency of heralded photons, and c success rate of the protocol for the fraction of the runs with detected heralded photon n
g
c
Task A
6,692
0.452 ± 0.010
0.966 ± 0.003
Task B
18,169
0.471 ± 0.006
0.858 ± 0.004
Experimental multipartner quantum communication complexity
5 Generalization In the previous sections we have demonstrated how a specific type of multiparty quantum communication tasks, namely two CCPs, can be solved efficiently without the usage of entangled states. Note, our method is not restricted to this particular kind of problem. In fact, it gives a generic prescription how to simplify any multiparty quantum communication protocol that employs the resource of entangled GHZ states. These entanglement based protocols use a non-local quantum phase, which can be operated on locally by each party, and the particular correlations between local measurement results, typical for the GHZ state. Alternatively, the single-qubit protocols use a quantum phase of a qubit to sequentially encode local information as it flies by the parties towards the last party, who performs a suitable measurement. The expectation value of such a measurement has the same structure as the correlation function of a GHZ state. As an example, we want to point out at another protocol: secret sharing. It is an application, in which a secret message is split in a way that its reconstruction requires the collaboration of the participating partners. Splitting the message such that a single person is not able to read it is a common task in information processing and especially in high security applications. A solution for this problem and its generalization including several variations is provided by classical cryptography and consists of splitting the message using mathematical algorithms and distributing the resulting pieces to two or more legitimate users by classical communication. As generally known, all methods of classical communication currently used are susceptible to eavesdropping attacks. In contrast, the usage of quantum resources can lead to unconditionally secure communication (e.g. Gisin et al. 2002; Ekert 1991), consequently protocols introducing quantum _ cryptography to secret sharing were proposed (Zukowski et al. 1998; Hillery et al. 1999; Cleve 1999; Karlsson et al. 1998). In these protocols, a shared GHZ-state allows the information splitting and the eavesdropper protection simultaneously. The idea, which sounds impressively simple and admittedly ingenious on the first glance is again very demanding from an experimental point of view. Indeed, due to lack of efficient multiphoton sources an experimental demonstration of secret sharing requires significant resources for comparably low rates (Gaertner et al. 2007). In (Schmid et al. 2005) we proposed, and realized a protocol for N ? 1 parties in which only sequential single qubit communication between them is used instead of a multipartite GHZ state distribution. The new protocol
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requires only single qubits and is thus much more scalable with respect to the number of participating parties. These gains enabled the experimental demonstration of our protocol for six parties. We were able to establish between these parties a key of 982 secret bits with an quantum bit error rate of 2.34 %. The scheme involved essentially the same interferometric setup as the one used for the communication complexity tasks described here. To our knowledge this is the first experimental implementation of a full protocol for secret sharing and by far the highest ever reported number of participants in any quantum information processing task.
6 Summary We have proven and experimentally demonstrated the superiority of quantum communication over its classical counterpart for distributed computational tasks by solving two examples of CCPs. For nontrivial CCPs, where the input from all the partners is required in order to obtain a non-random final result, the best classical fidelity goes exponentially to 0 with increasing number N of partners. Yet, the fidelity stays constant and is independent of N for our single qubit assisted protocols. In our experimental realization we have reached higherthan-classical performance even when including all experimental imperfections of state-of-the-art technologies. That means, for the comparison we count every qubit emitted by the source and consequently manipulated by the partners, regardless of whether it finally was detected and thus useful for the protocol. The efficiency of the set-up together with the exponentially increasing advantage of the protocol makes the quantum solution superior to any classical protocol. Thus, by successfully performing fair and real comparison with the classical scenario with present-day technology we clearly illustrate the potential of the implemented scheme in real applications of multi-party quantum communication. Most importantly, our method gives a generic prescription how to simplify multi-party quantum communication protocols employing the resource of entangled GHZ states. Thus, our single qubit concept will increase the applicability for all the many other multiparty protocols relying on the particular features of quantum correlation functions. Acknowledgements This work was supported by the DFG, EU-FET (RamboQ, IST-2001-38864), Marie-Curie program and _ was supported by the VI DAAD/KBN exchange program. M.Z. Framewoerk EU programmes QAP and SCALA as well as by Wenner Gren Foundations.
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References Bennett CH, Brassard G (1984) Quantum cryptography: public key distribution and coin tossing. In: Proceedings of IEEE International conference on computers, systems and signal processing. IEEE, New York, p 175–179 _ Brukner Cˇ, Zukowski M, Pan J-W, Zeilinger A (2004) Bell’s inequalities and quantum communication complexity. Phys Rev Lett 92:127901 _ Brukner Cˇ, Zukowski M, Zeilinger A (2002) Quantum communication complexity protocol with two entangled qutrits. Phys Rev Lett 89:197901 Buhrman H, Cleve R, van Dam W (2001) Quantum entanglement and communication complexity. Siam J Comput 30:1829–1841; [eprint: quant-ph/9705033] Buhrman H, Cleve R, Wigderson A (1998) Quantum vs. classical communication and computation. In: Proceedings of the 30th annual ACM symposium on theory of computing. ACM Press, New York, p 63–68 Buhrman H, van Dam W, Høyer P, Tapp A (1999) Multiparty quantum communication complexity. Phys Rev A 60:2737–2741 Cabello A, Lo´pez-Tarrida J (2005) Proposed experiment for the quantum ‘‘Guess My Number’’ protocol. Phys Rev A 71:020301(R) Cleve R, van Dam W, Nielsen M, Tapp A (1999) Quantum entanglement and the communication complexity of the inner product function. Lecture notes in computer science, vol. 1509. Springer, London, p 61–74 Cleve R, Gottesmann D, Lo H-K (1999) How to share a quantum secret. Phys Rev Lett 83:648 Ekert AK (1991) Quantum cryptography based on Bell’s theorem. Phys Rev Lett 67:661 Gaertner S, Kurtsiefer C, Bourennane M, Weinfurter H (2007) Experimental demonstration of four-party quantum secret sharing. Phys Rev Lett 98:020503 Galva˜o EF (2002) Feasible quantum communication complexity protocol. Phys Rev A 65:012318 Gisin N, Ribordy G, Tittel W, Zbinden H (2002) Quantum cryptography. Rev Mod Phys 74:145
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P. Trojek et al. Hardy L, van Dam W (1999) Quantum communication using a nonlocal Zeno effect. Phys Rev A 59:2635–2640 Hillery M, Buzˇek V, Berthiaume A (1999) Quantum secret sharing. Phys Rev A 59:1829 Holevo AS Bounds for the quantity of information transmitted by a quantum communication channel. Probl Peredachi Inf 9:3–11 (1973) [transl: Probl Inf Transm 9:177–183 (1973)] Horn RT, Babichev SA, Marzlin K-P, Lvovsky AI, Sanders BC (2005) Single-qubt optical quantum fingerprinting. Phys Rev Lett 95:150502 Karlsson A, Koashi M, Imoto N (1998) Quantum entanglement for secret sharing and secret splitting. Phys Rev A 59:162 Kushilevitz E, Nisan N (1997) Communication complexity. Cambridge University Press, Cambridge Raz R (1999) Exponential separation of quantum and classical communication complexity. In: Proceedings of the 31th annual ACM symposium on theory of computing. ACM Press, New York, pp 358–367 _ Schmid C, Trojek P, Bourennane M, Kurtsiefer C, Zukowski M, Weinfurter H (2005) Experimental single qubit quantum secret sharing. Phys Rev Lett 95:230505 _ Trojek P, Schmid C, Bourennane M, Brukner Cˇ, Zukowski M, Weinfurter H (2005) Experimental quantum communication complexity. Phys Rev A 72:050305(R) Xue P, Huang Y-F, Zhang Y-S, Li C-F, Guo G-C (2001) Reducing the communication complexity with quantum entanglement. Phys Rev A 64:032304 Yao AC-C (1979) Some complexity questions related to distributed computing. In: Proceedings of the 11th annual ACM symposium on theory of computing. ACM Press, New York, pp 209–213 Zhang J, Bao X-H, Chen T-Y, Yang T, Cabello A, Pan J-W (2007) Experimental quantum ‘‘Guess my Number’’ protocol using multiphoton entanglement. Phys Rev A 75:022302 _ Zukowski M (1993) Bell theorem involving all settings of measuring apparatus. Phys Lett A 177:290–296 _ Zukowski M, Zeilinger A, Horne MA, Weinfurter H (1998) Quest for GHZ states. Acta Phys Pol 93:187
Experimental multipartner quantum communication complexity employing just one qubit Pavel Trojek • Christian Schmid • Mohamed Bourennane ˇ aslav Brukner • Marek Zukowski _ • C Harald Weinfurter
•
Published online: 5 January 2013 Ó Springer Science+Business Media Dordrecht 2013
Abstract Most proposals for quantum solutions of information-theoretic problems rely on the usage of multi-partite entangled states which are still difficult to produce experimentally with current state-of-the-art technology. Here, we analyze a scheme to simplify a particular kind of multiparty communication protocols for the experiment. We prove that the fidelity of two communication complexity protocols, allowing for an N - 1 bit communication, can be exponentially improved by N - 1 (unentangled) qubit communication. Taking into account, for a fair comparison, all inefficiencies of state-of-the-art set-up, the experimental implementation for N = 5 outperforms the best classical protocol, making it the candidate for multi-party quantum communication applications. Keywords Communication complexity Quantum communication Quantum information Parametric down conversion P. Trojek (&) C. Schmid H. Weinfurter Ludwig-Maximilians-Universita¨t, 80799 Mu¨nchen, Germany e-mail: [email protected] P. Trojek C. Schmid H. Weinfurter Max-Planck-Insititut fu¨r Quantenoptik, 85748 Garching, Germany M. Bourennane Physics Department, Stockholm University, 10691 Stockholm, Sweden Cˇ. Brukner Institut fu¨r Experimentalphysik, Universita¨t Wien, Boltzmanngasse 5, 1090 Wien, Austria _ M. Zukowski Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gdanski, 80-952 Gdansk, Poland
1 Introduction Quantum information science overcomes a number of barriers for conventional information transfer, cryptography and computation. However, so far the only quantum scheme that reached the stage of commercial application is quantum key distribution (Bennett and Brassard 1984). All other quantum information processes, to reach this stage, require the simplification of the quantum-assisted protocols and/or development of new experimental techniques for controlling and processing of quantum information. In particular multi-party quantum communication protocols require multipartite entangled states, which are very difficult to be produced with current methods and moreover suffer from a high noise. In the domain of photonic qubits, the yield of multi-partite entangled states from, currently used, spontaneous parametric down conversion (SPDC)schemes decreases exponentially with the number of partners. It was shown recently that entanglement is not the only non-classical resource endowing the quantum multiparty information processing its power. Instead, even the sequential communication and transformation of a single qubit can be sufficient (Galva˜o 2002; Buhrman et al. 1998; Raz 1999). This breakthrough makes multi-party communication tasks feasible, and remarkably, technologically comparable to quantum key distribution as we are going to show in this contribution. One of the first significant advantages of introducing quantum phenomena to communication problems was recognized in the field of communication complexity problems (CCP’s) (Yao 1979). There the goal is to find the minimum amount of information, which parties, performing some local computations, have to exchange in order to accomplish some globally defined goal. For example, in a two-party CCP, the parties P 1 and P 2 receive each an n-bit
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string X1 and X2 with the common goal to compute the value of a given function T(X1,X2). Before distribution of the strings, the parties are allowed to communicate freely and prepare themselves jointly in any way, e.g., by sharing classically correlated random variables, agreeing on a common strategy etc. However, once received, they must communicate as little as possible. The question is how many bits of information exchange then is enough to compute T(X1,X2) (Buhrman et al. 1998, 1999; Cleve et al. 1999), or alternatively how high is the probability to reach the correct value of T(X1,X2) under a specific communication restriction (Buhrman et al. 2001; Brukner et al. 2002, 2004; Galva˜o 2002). The purpose of such studies is making distributed computations efficient in terms of communication, and thus speeding them up. Besides, such an abstract problem finds its practical use in many other applications, including the design of Very Large Scale Integrated circuits or study of data structures (Kushilevitz and Nisan 1997). Quantum protocols involving multiparty entangled states were shown to be superior to classical protocols for a number of CCPs (Buhrman et al. 2001; Cleve et al. 1999; Buhrman et al. 1999; Hardy and van Dam 1999). As a criterium of the advantage, the violation of a Bell inequality, related with the task of the protocol, can be used (Brukner et al. 2002, 2004). Despite many theoretical results, with exception of the experiment that we report in the paper, no experiment showing a genuine quantum advantage in solving CCPs has been reported; see also feasibility studies (Galva˜o 2002; Cabello and Lo´pez-Tarrida 2005). The reason is the above mentioned problem concerning the generation of multi-partite entangled states. The only laboratory demonstrations so far, corrected the measured results for the experimental inefficiencies (Xue et al. 2001; Horn et al. 2005; Zhang et al. 2007), and thus did not provide the conclusive and unambiguous practical manifestation of the superiority of quantum-assisted protocols in distributed computations. Here we show that, for solving CCPs with restricted communication, the usage of entangled states is not required and that single qubit assisted protocols are indeed superior over the corresponding classical ones. Their superiority may increase even exponentially with the number of partners. We review the experimental results obtained in (Trojek et al. 2005), where we could demonstrate by using parametric down-conversion as a source of heralded single qubits that quantum protocols solve two CCPs more efficiently, even with the limited detection efficiency inherent in any real experiments. We focus in the present work on the underlying theory and present the theoretical study which includes the explicit derivation of the upper classical bounds on success probability for the two examples of CCPs. Thereby we provide a rigorous proof of the exponential separation of
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quantum/classical solutions in terms of their efficiency. Finally we point out that our single qubit approach is generic for the solution of any communication protocol relying on the correlation function of GHZ states.
2 Theory 2.1 Communication complexity problems Let us introduce the two CCPs which we will analyze here and later on show their experimental implementations. The first one, problem A, is the so-called modulo-4 sum problem. It was first defined for three parties by Buhrman et al. (2001) and later generalized to N parties in (Buhrman et al. 1999). The problem in the setting with restricted communication is stated as follows: Imagine N separated partners P 1 ; . . .; P N . Each of them receives a two-bit input string Xk ; ðXk ¼ 0; 1; 2; 3; k ¼ 1; . . .; NÞ: The Xks are distributed such that their sum is even, P i.e., ( Nk=1 Xk)mod 2 = 0. No partner has any information whatsoever on the values received by the others. Next, the partners communicate with the goal that one of them, say P N ; can tell whether the sum modulo-4 of all inputs is equal zero or two. That is, P N should announce the value of a dichotomic, i.e. Pof values ±1, function TðX1 ; . . .; XN Þ given by TA = 1 ( Nk=1 Xkmod 4). The total amount of communication is restricted to only N - 1 bits (classical scenario). The partners can freely choose a communication protocol as long as it does not depend on input data. Such a dependence would imply a violation of the communication restriction. (E.g. they can choose between sequential communication from one to the other, or any arbitrary tree-like structure ending at the last party P N :) An alternative description of the task A, simplifying the calculation of the maximum classical success probability and making the connection with the task B more visible, can be introduced. It puts the probability distribution for local data as ! N 1 pX pA ðX1 ; . . .; XN Þ ¼ 2N1 cos ð1Þ Xk ; 2 2 k¼1 and the global task function as ! N pX TA ðX1 ; . . .; XN Þ ¼ cos Xk : 2 k¼1
ð2Þ
Problem B has a similar structure as A, but now N real numbers X1 ; . . .; XN 2 ½0; 2pÞ with probability density pB ðX1 ; . . .; XN Þ ¼
1 4ð2pÞN1
jcosðX1 þ þ XN Þj
ð3Þ
are distributed to the partners. Their task is to compute whether cosðX1 þ . . . þ XN Þ is positive or negative, i.e. to give the value of the dichotomic function
Experimental multipartner quantum communication complexity
"
N X
TB ¼ S cos
21
!# Xk
;
ð4Þ
k¼1
where S(x) = x/|x|. The communication restriction is the same as for problem A, i.e., only N - 1 bits are allowed to be exchanged. 2.2 Optimal classical protocol To find the best performing protocols for these two CCPs, it is convenient to first rewrite the random inputs Xks. For the task A we put Xk = (1 - yk) ? xk, where yk 2 f1; 1g; xk 2 f0; 1g: For the task B we write Xk = p (1 yk)/2 ? xk, with yk 2 f1; 1g; xk 2 ½0; pÞ: Accordingly, the task function T can now be reformulated in a common Q form: T ¼ f ðx1 ; . . .; xN Þ Nk¼1 yk and pðX1 ; . . .; XN Þ ¼ 2N p0 ðx1 ; . . .; xN Þ: This reformulation enables one to adopt the mathematical formalism developed for entanglement-assisted communication complexity models in (Brukner et al. 2002, 2004). Specifically, for the task A P we have f ¼ fA ¼ cosðp2 Nk¼1 xk Þ with p0 ¼ p0A ¼ 2Nþ1 P j cos p2 Nk¼1 xk j; and for the task B; f ¼ fB ¼ P P S½cosð Nk¼1 xk Þ with p0 ¼ p0B ¼ 21 pNþ1 j cosð Nk¼1 xk Þj. We note that the dichotomic variables yk are not restricted by the probability distributions, p, for the Xks. Thus, they are completely random. Furthermore, since T is proportional to the product of all yks, the answer eN = ± 1 of P N is completely random with respect to T, if it does not depend on every yk. Thus, an unbroken communication structure is necessary: the information from all N - 1 partners must directly or indirectly reach P N : Due to the restriction to N 1 bits of communication each of the partners, P k ; where k ¼ 1; . . .; N 1; sends only a one-bit message, which for convenience will be denoted as ek = ±1. The task function T as well as the answer eN of P N takes only two values, ±1. If the answer is correct, then T = eN and thus TeN = 1; otherwise, TeN = -1. Therefore, following (Brukner et al. 2004] we can quantify the average P success of the protocol with the fidelity F ¼ X1 ;...;XN pTeN ; or equivalently F¼
X 1 p0 ðx1 ; . . .; xN Þf ðx1 ; . . .; xN Þ N 2 x ;...;x ¼0;1 1
N
X
N Y
ð5Þ
yk eN ðx1 ; . . .; xN ; y1 ; . . .; yN Þ
directly from partners P i1 ; . . .; P i‘ : That is, eN ¼ eðxN ; yN ; ei1 ; . . .; ei‘ Þ: Let us fix xN, and treat e as a function ex_N of the remaining ‘ ? 1 dichotomic variables, yN ; ei1 ; . . .; ei‘ : Such functions can be represented as 2‘?1 dimensional vectors. These, in turn, can be expanded with the use of an orthogonal basis given by Vjj1 ...j‘ Q ðyN ; ei1 ; . . .; ei‘ Þ ¼ yjN ‘k¼1 ejikk ; where j; j1 ; . . .; j‘ ¼ 0; 1: Therefore, one can express ex_N in the following way: cjj1 ...j‘ ðxN ÞyjN
j;j1 ;...;j‘ ¼0;1
‘ Y
ejikk ;
ð6Þ
k¼1
where cjj1 ...j‘ ðxN Þ ¼
1 2‘þ1 y
X
exN V jj1 ...j‘ :
ð7Þ
N ;ei1 ;...;ei‘ ¼1
Since jexN j ¼ jVjj1 ...j‘ j ¼ 1; one has jcjj1 ...j‘ ðxN Þj 1: We put P 0 the expansion to Eq. (5). As, y_N= ± 1yNyN = 0, and P 0 y_k = ± 1ykek = 0, only the term with j; j1 ; . . .; j‘ ¼ 1 in expansion (6) can give a non-zero contribution to Fc. Thus, without changing the result of Eq. (5), eN in (5) can Q be replaced by a function e0N ¼ yN cN ðxN Þ ‘k¼1 eik : where cN(xN) stands for c11...1 ðxN Þ: Next, notice that, e.g., ei_1, which is in the formula for e0 N, can depend only on xi_1, yi_1 and the messages obtained by P i1 from a subset of partners: ep1 ; . . .; epm (this set does not contain any ei_k). In analogy with (6), ei_1, for a fixed xi_1, can be expanded in terms of orthogonal basis functions: ei1 ¼
X
c0jj1 ...jm ðxi1 Þyji1
j;j1 ;...;jm ¼0;1
m Y
ejpkk :
ð8Þ
k¼1
Again, jc0jj1 ...jm ðxi1 Þj 1: If one puts this into e0 N one obtains for the fidelity 1 X Fc ¼ N2 gðx1 ; . . .; xN ÞcN ðxN Þci1 ðxi1 Þ 2 x1 ;...;xN ð9Þ m ‘ X Y Y Y yk epr eik ; y0 k6¼N;i1
r¼1
k¼2
P where g ¼ p0 f ; ci1 ðxi1 Þ ¼ c011...1 ðxi1 Þ; and y’ represents summation over y1 ; . . .; yi1 1 ; yi1 þ1 ; . . .; yN1 ¼ 1: Note that each message appears in the product only once. We continue this procedure of expanding the messages, till it halts (i.e., till we reach the level of those partners who do not receive any messages). The end result is
y1 ;...;yN ¼1 k¼1
The probability of success reads P = (1 ? F)/2. For the problem B integrations replace summations Rp P ð xk ! 0 dxk Þ: In any classical protocol the answer eN given by P N can depend on yN, xN, and on the messages, ei1 ; . . .; ei‘ ; received
X
e xN ¼
Fc ¼
X x1 ;...;xN
gðx1 ; . . .; xN Þ
N Y
cn ðxn Þ;
ð10Þ
n¼1
with |cn(xn)| B 1. Since Fc in Eq. (10) depends on the product of local functions cn(xn), it has to be bounded from above, i.e., jFc j BðNÞ; where BðNÞ is the maximum
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classical fidelity. The reasoning is similar to the one leading to Bell inequalities; (as a matter of fact the inequality jFc j BðNÞ; where Fc is defined in (10), is an algebraic version of a general Bell inequality for N observers (Brukner et al. 2002, 2004]). The extrema of Fc are at the limiting values cn(xn) = ± 1, because Fc is linear in every cn(xn). It becomes apparent now that the class of protocols, in which the partners P 1 to P N1 calculate ek = ykck(xk), with ck(xk) = ± 1, and send the Q result, encoded in ek, to P N ; who puts eN ¼ Nk¼1 yk ck ðxk Þ; contains an optimal protocol of fidelity BðNÞ: Let us calculate the specific forms of classical fidelity bounds BðNÞ for both problems. With the use of formulas (1) and (2) we obtain for the problem A the following form of Fc,A:
Fc;B ¼
1 2pN1
Zp
Zp dx1 . . .
dxN cosðx1 þ . . .
0" 0 !# N N Y X xk ck ðxk Þ: þ xN Þ S cos k¼1
ð17Þ
k¼1
Since jyj S½y ¼ y and |ck(xk)| = 1, one can derive the _ following inequality (Zukowski 1993): ! p p Z Z N N Y X dx1 . . . dxN cos xk ck ðxk Þ 2N : ð18Þ 0
0
k¼1
k¼1
ð12Þ
The derivation goes as follows. One expands the cosine function, and notices that the intergrations over xi are always Rp of the following form 0 dxi bðxi Þci ðxi Þ; where b(xi) is sin xi or Rp cos xi : It is easy to notice that if 0 dxi ci ðxi Þ sin xi ¼ Bi sin ai Rp then one must have 0 dxi ci ðxi Þ cos xi ¼ Bi cos ai ; where ai Rp is a certain angle, and Bi ¼ maxb 0 dxi sin ðxi þ bÞci ðxi Þ: Of course Bi B 2. Putting all that into the left hand side of the inequality (18), we get straightforwardly the right hand side bound. Therefore, the classical bound BðNÞ for the task B involving any number of parties N is given by N1 1 2 N BðNÞ ¼ maxðFc;B Þ ¼ N1 2 ¼ : ð19Þ 2p p
Since ck are dichotomic functions of spectrum ±1, the possible values of (12) are p pffiffiffi p 1 i ¼ 2 exp i þ in ; ð13Þ 4 2
In both cases the fidelity decreases exponentially with the number of parties N. For the task A we found Fc,A B 2-K?1, where K = N/2 and K = (N ? 1)/2 for even and odd numbers of parties, respectively. This analytic result confirms the numerical simulations of (Galva˜o 2002) for small N. For the task B we derived Fc,B B (2/p)N-1.
Fc;A
! N N Y pX ¼ N1 cos xk ck ðxk Þ 2 2 k¼1 x1 ;...;xN k¼1 " # N X p Y 1 ¼ N1 Re exp ixk ck ðxk Þ : 2 2 k¼1 xk 1
X
ð11Þ
Each of the numbers x1 ; . . .; xN takes only the values zero and one, so the sum in (11) reduces to X xk
p exp ixk ck ðxk Þ ¼ ck ð0Þ þ ick ð1Þ: 2
where n is an integer. Considering an even number N of parties, i.e. N = 2 K, K being an integer C1, the expression in (11) is maximized by using pairs of conjugate values (13). Thus the maximum is given by BðN ¼ 2KÞ ¼ maxðFc;A Þ " # K pffiffiffi p p Y 1 ppffiffiffi p 2 exp i in 2 exp i þ in ¼ 2K1 Re 2 4 2 4 2 i¼1 ð14Þ and one can easily show that the bound for even number of parties is BðN ¼ 2KÞ ¼ 2Kþ1 :
ð15Þ
Analogously, for odd number of parties, i.e. N = 2K - 1, we can find the bound BðN ¼ 2K 1Þ ¼ 2Kþ1 : For the problem B, the fidelity Fc,B is given by
123
ð16Þ
2.3 Quantum protocol For the quantum protocols, we note that the Holevo (1973) bound limits the information storage capacity of a qubit to no more than one classical bit. Thus, we must now restrict the communication to N - 1 qubits, or alternatively, to N - 1-fold exchange of a single qubit. The optimal solution of the task A starts with a qubit pffiffiffi in the state j w0 i ¼ 1= 2ðj 0iþ j 1iÞ: Parties then sequentially act on the qubit with the phase-shift transformation j 0ih0 j þeipXk =2 j 1ih1 j; in accordance with their local data Xk. After all N phase shifts the state takes the form: PN 1 j wN i ¼ pffiffiffi j 0i þ eipð k¼1 Xk Þ=2 j 1i : ð20Þ 2 Since the sum over Xk is even, the phase factor PN eipð k¼1 Xk Þ=2 is equal to the dichotomic function TA to be computed. Thus, a measurement of the qubit in the basis
Experimental multipartner quantum communication complexity
23
pffiffiffi ðj 0i j 1iÞ= 2 reveals the value of TA with fidelity Fq,A = 1, that is, always correctly. Task B starts also with a qubit in the state j w0 i: Each party performs according to his/her local data a unitary transformation j 0ih0 j þeiXk j 1ih1 j; leading to PN 1 ð21Þ j wN i ¼ pffiffiffiðj 0i þ ei k¼1 Xk j 1iÞ: 2 The last party makes the same measurement as in the task pffiffiffi A. The probability for the detection of state 1= 2ðj 0i j 1iÞ; which we associate with the result r = ±1, is given P by PðÞ ¼ ½1 cosð Nk¼1 Xk Þ=2: The expectation value for the final answer eN = r is E = P(? ) - P(-), and P reads cosð Nk¼1 Xk Þ: The fidelity of eN, with respect to TB is Fq;B ¼
Z2p
Z2p dX1 . . .
0
dXN pB ðX1 ; . . .; XN ÞTB ðX1 ; . . .; XN Þ
0
EðX1 ; . . .; XN Þ:
ð22Þ
With the actual forms of pB, TB, and E, one gets Fq,B = p/ 4, i.e., the protocol gives the correct value of TB with probability Pq,B = (1 ? p/4)/2 & 0.892. Thus, the simple, qubit assisted quantum protocols are proven to be significantly more efficient than any classical protocols for both problems. Expressing the efficiency of protocols in terms of fidelities even an exponential separation for classical/quantum scenario is obtained, i.e. the ratio Fc/Fq decreases exponentially with N! Moreover, Fq for both problems are independent of N.
3 Experiment We implemented the quantum protocols for N = 5 parties (Trojek et al. 2005), using a heralded single photon as the carrier of the qubit communicated sequentially by the partners. The qubit was encoded in the polarization, so that the computational basis, ‘‘0‘‘ and ‘‘1’’, corresponds to horizontal H and vertical V linear polarization, respectively. The data Xk of each party was encoded on the qubit via a phase shift, using birefringent materials. The last pffiffiffi party performed a measurement in the basis 1= 2ðj Hi j ViÞ to obtain the answer eN. The experimental set-up is shown in Fig. 1. Time-correlated photon pairs are produced via non-collinear SPDC process in a 2-mm-long BBO crystal pumped by a blue LD (kp = 402.5 nm) with the output optical power of about 10 mW. The detection of one photon by the trigger detector DT heralds the existence of the other one used in the protocol. The narrow gate window of 4 ns for coincidence detection between these two photons, along with the singlecount rates of & 140,000 s-1 at the detectors D? and D-,
warrant that the recorded data are due to single photons; the probability of having more than one photon per heralding signal and during time corresponding to the gate window is estimated to 2 9 10-3, assuming the given parameters and the Poissonian photon-number statistics of SPDC photons. Since a type II degenerate phase-matching scheme is used, the emitted photon pairs at k = 805 nm (Dk 6 nm) are orthogonally polarized. Filtering of the vertical polarization for the trigger photons, ensures that the protocol photon has horizontal polarization initially. A half-wave plate (HWP1) transforms the qubit to the initial state pffiffiffi 1= 2ðj Hiþ j ViÞ as required by the optimal quantum protocol. The individual phase shifts of parties are implemented using a 200 lm thick Yttrium-Vanadate (YVO4) birefringent uniaxial crystals (Ci). The crystals, cut with their optic axes parallel to the surface, are aligned in such a way that H and V polarization states correspond to their normal modes. In this configuration the length difference between optical paths of horizontal and vertical polarization can be continuously tuned using motor-driven rotations of the crystals along their optic axes, thereby allowing to set any desired phase-shift from 0 to 2p independently from the incoming polarization state. The precision in applying an arbitrary phase shift was measured to be better than 0.02p over the full phase range. An additional YVO4 crystal (Ccomp, 1,000 lm long), aligned with its optic axis in the plane perpendicular to the direction defined by the optic axes of the previous crystals, is used to compensate dispersion effects. To analyze the polarization state of photons pffiffiffi in the basis ðj 0i j 1iÞ= 2; a half wave-plate (HWP2) at an angle of 22.5° followed by polarizing beam-splitter (PBS) is used. For a fair comparison between the quantum protocols the classical one-bit protocols, no heralded events are
RNG
RNG
RNG
RNG
RNG
Fig. 1 Experimental set-up for solving of qubit-assisted CCPs. Pairs of orthogonally polarized photons are emitted from a BBO crystal via the type-II SPDC process. The detection of one photon by the trigger detector DT indicates the existence of the protocol photon. The polarization state is prepared with a half-wave plate (HWP1) and a polarizer, placed in the trigger arm. According to the local data (X1 to X5), each of the parties introduces a phase-shift by the rotation of a birefringent YVO4 crystal (C1–C5). The last party performs the analysis of a photon-polarization state using a half-wave plate (HWP2) followed by a polarizing beam-splitter (PBS)
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discarded, even if the detection of the protocol photon fails. In such a case one can still guess the value of T, but with success rate of only 1/2. Therefore, a high detection efficiency g of the heralded photons, i.e. high coincidence/ single ratio for our set-up, is essential for an unambiguous demonstration of the superiority of the qubit-assisted protocol. E.g., in a realistic implementation of the five-party problem A, the efficiency of gJ0:33 was estimated to be sufficient to beat the optimum classical protocol, see (Galva˜o 2002). To minimize the cases with no detection of the photon, the yield of heralded photons was maximized by adopting an unbalanced SPDC scheme. We select a restricted spatial mode with well defined polarization of the trigger photons by coupling them into a single-mode fiber behind a polarizer, whereas no spatial filtering is performed on the protocol photons. As a result, we observed & 5000 trigger events per second at the detector DT with & 2,400 coincident events per second of protocol detections, i.e. an overall detection efficiency of g& 0.48, close to the limit given by the detector efficiency of our silicon APDs (measured to be about & 55 %). The protocols were run many times, to obtain sufficient statistics. Each run took about one second. It consisted of generating a set of pseudorandom numbers obeying the specific distribution, subsequent setting of the corresponding phase shifts, and opening detectors for a collection time window s. The limitation of communicating one qubit per run requires that only these runs, in which exactly one trigger photon is detected during s, are selected for the evaluation of the probability of success Pexp. To maximize the number of such runs, n, the length of s was optimized to 200 ls, assuming a Poissonian photon-number distribution of SPDC photons. We merely remark that essentially the same set-up could be used to solve the CCPs using bright polarized pulses. However, in such a case, a suitable polarization measurement of the pulses reveals all the encoded input data of any party: two bits for the task A, and arbitrarily many for the task B. Thus, the communication restriction to N - 1 bits is violated. Attenuation of the pulses to the single-photon level does not help either. The efficiency of the protocol is significantly lowered in such a case, due to many nondetection events, forcing one to guess the answer most of the time.
4 Results In order to determine the probability of success from the data acquired during the runs we have to distinguish the following two cases. First, the heralded photon is detected, which happens with probability g, given by the
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coincidence/single ratio. Then, the answer eN can be based on the measurement result. However, the answer is correct only with a probability c, due to experimental imperfections in the preparation of the initial state, the setting of the desired phase shifts, and the polarization analysis. This must be compared with the theoretical limits given by Pq,A and Pq,B for the task A and B, respectively. Second, with the probability 1 - g the detection of the heralded photon fails. Forced to make a random guess, the answer is correct in half of the cases. This leads to an overall success probability Pexp = gc ? (1 - g)0.5, or a fidelity of Fexp = g(2c - 1). Due to a finite measurement sample, our experimental results for the success probability are distributed around the value Pexp as shown in Fig. 2 for both tasks. [tb] The width of the distribution is interpreted as the error in the experimental success probability. For the task A we obtain a quantum success probability of Pexp,A = 0.711 ± 0.005. The bound Pc,A = 5/8 of the optimal classical protocol for N = 5 parties is violated by 17 standard deviations. For the task B we reached Pexp,B = 0.669 ± 0.003, whereas the classical bound is Pc,B & 0.582. The violation is by 29 standard deviations. Expressing the final results in terms of fidelities, we obtain Fexp,A = 0.421 ± 0.010 for the task A, and Fexp,B = 0.337 ± 0.006 for the task B. This must be compared to the performance of the best classical protocol for N = 5 parties reaching Fc,A = 0.25 for A and Fc,B&0.164 for B. Table 1 summarizes the relevant experimental parameters for both tasks.
(a)
(b)
Fig. 2 Histograms of measured quantum success probabilities a for the task A and b for task B. The bounds for optimum classical protocols are displayed, too
Table 1 Relevant experimental parameters in the implementation of communication complexity problems A and B: n number of correct runs, g overall detection efficiency of heralded photons, and c success rate of the protocol for the fraction of the runs with detected heralded photon n
g
c
Task A
6,692
0.452 ± 0.010
0.966 ± 0.003
Task B
18,169
0.471 ± 0.006
0.858 ± 0.004
Experimental multipartner quantum communication complexity
5 Generalization In the previous sections we have demonstrated how a specific type of multiparty quantum communication tasks, namely two CCPs, can be solved efficiently without the usage of entangled states. Note, our method is not restricted to this particular kind of problem. In fact, it gives a generic prescription how to simplify any multiparty quantum communication protocol that employs the resource of entangled GHZ states. These entanglement based protocols use a non-local quantum phase, which can be operated on locally by each party, and the particular correlations between local measurement results, typical for the GHZ state. Alternatively, the single-qubit protocols use a quantum phase of a qubit to sequentially encode local information as it flies by the parties towards the last party, who performs a suitable measurement. The expectation value of such a measurement has the same structure as the correlation function of a GHZ state. As an example, we want to point out at another protocol: secret sharing. It is an application, in which a secret message is split in a way that its reconstruction requires the collaboration of the participating partners. Splitting the message such that a single person is not able to read it is a common task in information processing and especially in high security applications. A solution for this problem and its generalization including several variations is provided by classical cryptography and consists of splitting the message using mathematical algorithms and distributing the resulting pieces to two or more legitimate users by classical communication. As generally known, all methods of classical communication currently used are susceptible to eavesdropping attacks. In contrast, the usage of quantum resources can lead to unconditionally secure communication (e.g. Gisin et al. 2002; Ekert 1991), consequently protocols introducing quantum _ cryptography to secret sharing were proposed (Zukowski et al. 1998; Hillery et al. 1999; Cleve 1999; Karlsson et al. 1998). In these protocols, a shared GHZ-state allows the information splitting and the eavesdropper protection simultaneously. The idea, which sounds impressively simple and admittedly ingenious on the first glance is again very demanding from an experimental point of view. Indeed, due to lack of efficient multiphoton sources an experimental demonstration of secret sharing requires significant resources for comparably low rates (Gaertner et al. 2007). In (Schmid et al. 2005) we proposed, and realized a protocol for N ? 1 parties in which only sequential single qubit communication between them is used instead of a multipartite GHZ state distribution. The new protocol
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requires only single qubits and is thus much more scalable with respect to the number of participating parties. These gains enabled the experimental demonstration of our protocol for six parties. We were able to establish between these parties a key of 982 secret bits with an quantum bit error rate of 2.34 %. The scheme involved essentially the same interferometric setup as the one used for the communication complexity tasks described here. To our knowledge this is the first experimental implementation of a full protocol for secret sharing and by far the highest ever reported number of participants in any quantum information processing task.
6 Summary We have proven and experimentally demonstrated the superiority of quantum communication over its classical counterpart for distributed computational tasks by solving two examples of CCPs. For nontrivial CCPs, where the input from all the partners is required in order to obtain a non-random final result, the best classical fidelity goes exponentially to 0 with increasing number N of partners. Yet, the fidelity stays constant and is independent of N for our single qubit assisted protocols. In our experimental realization we have reached higherthan-classical performance even when including all experimental imperfections of state-of-the-art technologies. That means, for the comparison we count every qubit emitted by the source and consequently manipulated by the partners, regardless of whether it finally was detected and thus useful for the protocol. The efficiency of the set-up together with the exponentially increasing advantage of the protocol makes the quantum solution superior to any classical protocol. Thus, by successfully performing fair and real comparison with the classical scenario with present-day technology we clearly illustrate the potential of the implemented scheme in real applications of multi-party quantum communication. Most importantly, our method gives a generic prescription how to simplify multi-party quantum communication protocols employing the resource of entangled GHZ states. Thus, our single qubit concept will increase the applicability for all the many other multiparty protocols relying on the particular features of quantum correlation functions. Acknowledgements This work was supported by the DFG, EU-FET (RamboQ, IST-2001-38864), Marie-Curie program and _ was supported by the VI DAAD/KBN exchange program. M.Z. Framewoerk EU programmes QAP and SCALA as well as by Wenner Gren Foundations.
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