Carrying an arbitrarily large amount of information using a single quantum particle
Li-Yi Hsu, Ching-Yi Lai, You-Chia Chang, Chien-Ming Wu, Ray-Kuang Lee
CCarrying an arbitrarily large amount of information using a single quantum particle
Li-Yi Hsu, Ching-Yi Lai, ∗ You-Chia Chang, Chien-Ming Wu, and Ray-Kuang Lee Department of Physics, Chung Yuan Christian University, Chungli 32023, Taiwan Institute of Communications Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan Department of Photonics and Institute of Electro-Optical Engineering,National Chiao Tung University, Hsinchu 30010, Taiwan Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan (Dated: July 30, 2020)Theoretically speaking, a photon can travel arbitrarily long before it enters into a detector, result-ing a click. How much information can a photon carry? We study a bipartite asymmetric “two-waysignaling” protocol as an extension of that proposed by Del Santo and Daki´c. Suppose that Aliceand Bob are distant from each other and each of them has an n -bit string. They are tasked toexchange the information of their local n-bit strings with each other, using only a single photonduring the communication. It has been shown that the superposition of different spatial locationsin a Mach-Zehnder (MZ) interferometer enables bipartite local encodings. We show that, after thetravel of a photon through a cascade of n -level MZ interferometers in our protocol, the one of Aliceor Bob whose detector clicks can access the other’s full information of n -bit string, while the othercan gain one-bit of information. That is, the wave-particle duality makes two-way signaling possible,and a single photon can carry arbitrarily large (but finite) information. I. INTRODUCTION
Communication is a process of sending and receivingmessages from one party to another [1]. More precisely,communication is a physical process with physical infor-mation carriers transmitted without violating any phys-ical principle. For instance, electromagnetic waves usedin wireless communication are governed by Maxwell’sequations in classical physics. As a consequence of spe-cial relativity, faster-than-light communication is impos-sible. Also, the unavoidable energy consumption in theMaxwell’s demon and Landauer’s erasure indicates thatinformation is physical [2], and the link between thermo-dynamics and information has potential to deliver newinsights in physics and biology.The role of information in physics theory has been ex-tensively investigated. For example, it is proposed thatquantum theory can be derived and reconstructed frompurely informational principles [3–6]. The effect of uncer-tainty relation in information processing can be statedin terms of information content principle [7] and No-Disturbance-Without-Uncertainty principle [8]. Therein,a fundamental and interesting concern is the channel ca-pacity in communication. According to the no-signalingprinciple, there is no information gain without classi-cal or quantum communication; the transmission of themessage as the cause that increases the information. Itis well known that, in the dense-coding protocol, twobits of information can be carried in one qubit with pre-shared entanglement [9]. For the receiver to obtain n bits of information, at least a total of n qubits haveto be exchanged and at least n/ ∗ [email protected] tum physics, information causality states that one cannotgain more information than the number of bits sent viaclassical communication [17]. Note that the protocolsmentioned above are proposed for one-way communica-tion, and quantum entanglement as physical resource isinitially distributed between the sender and receiver.Photons as flying qubits are usually exploited in quan-tum communication. Given a photon as an informationcarrier, its particle-wave duality makes two-way commu-nication possible. Very recently a variant of the “guessyour neighbor’s input” game [18] was studied by Santoand Daki´c [19], which we call the SD game in this article.They proposed a protocol (SD protocol) to win the SDgame with certainty, while a classical strategy can winwith probability at most 50%. We review the SD gameas follows. Two distant agents Alice and Bob are giventwo input bits x, y ∈ { , } , respectively, which are drawnuniformly at random, and they are asked to output twobits a, b ∈ { , } , respectively. They win the game if bothof them output a bit that is equal to the other’s input(i.e., a = y and b = x ). A restriction here is that only aninformation carrier, classical or quantum, can be manip-ulated. Obviously, they cannot win with certainty usingsimply a classical information carrier since it can trans-mit a single bit of information within a specified timelimit. Using a photon, on the other hand, enables two-way signalling so that they can win the SD game withcertainty [20]. Notably, one of them can gain one bit of in-formation if no detector clicks. According to Renninger’snegative result experiment [21, 22] or the bomb-testingproblem [23], even if there is no interaction between thequantum object and the measuring device, one still learnsdefinite knowledge of the quantum state.The concept of SD protocol is explained as follows,in terms of the (level 1) optical implementation shownin Fig. 1. A photon is emitted from a referee sourceand then injected into the first beam splitter (BS 1) of a a r X i v : . [ qu a n t - ph ] J u l FIG. 1. Optical implementation of the SD protocol. Thephoton is initially injected into the level-1 MZ interferometer.After traveling through the first 50 /
50 beam splitter (BS1),the incident photon is half-reflected and half-transmitted ina coherent way. Alice and Bob can locally access the halvesdepicted in the red solid line and blue dash line, respectively.For the local encoding at level 1, Alice inserts the π -phasemodulator (PM A) if x = 0 and does nothing if x = 1;similarly, Bob inserts the π -phase modulators, (PM B) if y =0 and does nothing if y = 1. After the interference of thesetwo coherent halves at the second 50 /
50 beam splitter (BS2),the photon enters one of the two detectors [19].
Mach-Zehnder (MZ) interferometer. Consequently, thissingle photon is coherently superposed over two differentspatial locations. Hence the two local agents Alice andBob can each (i) perform local operations on the incom-ing parts of the photon as information encoding, and (ii)access a detector to detect the photon at a certain timewindow later. According to (i), Alice and Bob encodetheir bits in the phase of the photon before it reachesthe second beam splitter (BS2). With a delicate design,the parity of the two input bits completely determinesthe path of the photon leaving BS2. Consequently, oneknows with certainty which detector will detect this pho-ton while the other will detect nothing. For example, Al-ice’s (Bob’s) detector clicks if x = y ( x (cid:54) = y ) in the idealcase. Once Alice’s detector does not receive any pho-ton in a certain time window (no interaction between thequantum object and the measuring device), she knowsthat x (cid:54) = y and outputs bit a = x +1 mod 2. As a result,using the spacial superposition of a single photon, Aliceand Bob can communicate a total of two-bit informationwithin a specified time window and hence win the gamewith certainty. As opposed to this quantum communica-tion, to win the game with certainty using classical com-munication, the time-window would have been too shortto exchange two one-way classical communications [19].In this paper, we characterize the power of a single pho-ton as an information carrier. Our concerns are twofold:how much information a single photon can carry; andhow much information an agent can obtain even if aninteraction-free measurement occurs (no photon is de-tected by the detectors at hand). We will design a gen-eralized Santo and Daki´c (GSD) game and show thatusing one single photon, one can win the game with cer-tainty and learn a total of ( n + 1) bits of informationin an n -level GSD game, while one learns only n bits ofinformation by classical communication. When n is arbi- trarily large, this suggests that a single photon can carryan arbitrarily large amount of information. We wouldlike to mention that in a related work [24], Horvat andDaki´c showed that a single particle can be used to com-municate simultaneously with n parties and achieves theso-called genuine n -way signaling. Note that a photon asan information carrier here can be replaced by a quantumparticle whose coherence is under enough experimentalcontrol to exhibit coherence.The remainder of this paper is organized as follows:In Sec. II, we introduce the GSD game. The experimentsetup of the n -level circuit is proposed. We characterizeand then optimize the total information gains for Aliceand Bob. Several specific cases are studied. In Sec. III,we investigate the physics concerning the informationgains. Finally, in Sec. IV we estimate the performance ofthe GSD game in the physical realization. II. GENERALIZED SD GAMEA. Experimental setup
We consider a generalized Santo and Daki´c (GSD)game as follows. Alice and Bob are assigned two indepen-dent input strings x = x · · · x n , y = y · · · y n ∈ { , } n ,respectively, and they are asked to output bit strings a = a · · · a n and b = b · · · b n , respectively. They winthe game if (i) one of them can know the other’s inputstring and (ii) the other can gain at least one bit of infor-mation. Only a single information carrier is allowed forthe communication task within a specific time window.Equipped with a single photon in the GSD game, it willbe shown that there is a total of n + 1 bits of informationgain for Alice and Bob as a result of two-way signalingin a time window τ . However, if the information carrieris classical, they can exchange a total of at most n bitsof information in the same time window.First consider a two-level circuit as the extension ofthe SD protocol, as shown in Fig. 2. The two detectorsin Fig. 1 are replaced by MZ interferometers, followed byfour detectors. One of the four detectors will click accord-ing to the parities of ( x , y ) and ( x , y ). A ( k + 1)-levelcircuit can be constructed by (i) replacing the detectorsin the k -level circuit by the MZ interferometers, and (ii)putting 2 k +1 detectors at the output of the interferome-ters. Naturally, an n -level circuit for GSD game can berecursively extended.Our protocol for the SGD game is explained as followswith the experimental setup shown in Fig. 3, which canbe schematically depicted as a perfect n -level binary tree.A detector is placed at each leaf node, and a MZ inter-ferometer is placed at each parent node. According tothe input bits x i and y i , Alice and Bob perform phaseencoding by inserting a phase modulator (bit value = 0)or not (bit value = 1) into each of the 2 i MZ interfer-ometers at level i . Hence a single photon injected intothe root will travel through one of the 2 n light paths. FIG. 2. Left: The optical details of a two-level evolution cir-cuit. According to the local encoding at level 1, the leavingphoton at BS2 is injected into one of the two fibers (F1 andF2), and enters one of these two MZ interferometers at level 2.Similarly, Alice (Bob) inserts PMs into these two MZ inter-ferometers at level 2 if x = 1 ( y = 1) and does nothing if x = 0 ( y = 0). Right: The topological unfolding of the2-level circuit as a full 2-level binary tree. The nodes thereindenote the MZ interferometers, where the photon is spatiallysuperposed, and the directed edges between nodes indicatesthe possible travelling paths of the photon. Therein, after leaving a MZ interferometer at level k , thephoton goes either the even- k ( x k = y k ) path or odd- k ( x k (cid:54) = y k ) one, and then enters into a MZ interferometerat level ( k + 1). Note that there are 2 k − even- k and2 k − odd- k paths. Consequently, the photons completepath is determined by the parity relations of the n bitpairs ( x , y ) , . . . , ( x n = y n ) and finally flies into one of2 n detectors, D , . . . , D n , which are locally accessibleto either Alice or Bob. (Note that it is not necessarythat Alice and Bob have an equal number of detectors.)The local agent whose detector clicks can learn these n parities and hence knows the other’s n input bits exactly.Next we discuss the physical settings so that Alice andBob can exchange a single information carrier for a totalof n times. Let Alice and Bob be located at a distance d from each other, and, for simplicity, assume that aninformation carrier, classical or quantum, travels at thespeed c . Suppose that an information carrier carries onebit of information in classical one-way communication.So it takes time roughly nd/c for transmitting n bits ofinformation by a single carrier. On the other hand, let thelength of an odd- k or even- k path be δ for k < n . In otherwords, in Fig. 2, the photon travels a distance d betweenBS1 and BS2 is, and the length of F1 or F2 is δ . Inthe experiment setup, let δ (cid:28) d by choosing sufficientlylarge d ; however, such setup is not reflected from the scaleof our plots. Thus it takes time (( nd + ( n − δ ) /c ) toimplement our protocol in Fig. 3. As a result, we allowa specific time window τ such that (( nd + ( n − δ ) /c ) ≤ τ ≤ (( nd + ( n − δ ) /c ) + (cid:15) , where (cid:15) ≥ nd + ( n − δ ) /c ) + (cid:15) < ( n + 1) d/c . FIG. 3. The unfolding layout of the n -level circuit as aperfect n -level binary tree and the detectors. There are 2 i − MZ interferometers at level i . The photon is initially injectedinto the level-1 MZ interferometer. Then the parity of the bitpair ( x , y ) completely determines which one of the two level-2 MZ interferometers the photon will enter. Without loss ofgenerality, let the photon go to the right MZ interferometer atlevel 2 if x = y , and the left MZ interferometer, otherwise.More optical details are explained in Fig. 1. Similarly, theparity of ( x , y ) determines the next target interferometersat level 3. This process is continued for a cascade of n MZinterferometers. As a result, the light path of the photoncompletely depends on the n bit pairs ( x , y ), . . . , ( x n , y n ).Finally, the photon flies into one of the 2 n detectors, each ofwhich is held by Alice (A) or Bob (B). This choice of time window τ allows Alice and Bob toexchange a total of n + 1 bits of information (shown inthe next subsection) using our protocol, but this timewindow is not long enough so that a classical scheme canexchange only n bits of information. An example of n = 2 is illustrated in Fig. 4. FIG. 4. The time window in the n = 2 case. (a) Assumethat Alice holds the left two detectors ( D and D in Fig. 2).Alice and Bob each performs the local encoding operation for x at t = 0 and for y at t = ( d + δ ) /c , respectively. Finally,the photon flies into one of the detectors accessible to Aliceat time (2 d + δ ) /c ≤ τ ≤ (2 d + δ ) /c + (cid:15) . (b) In the samewindow τ , Alice and Bob can only exchange some x j and y i by classical communication. (c) To finish the same task asin (a) using a classical information carrier, it requires a timewindow 3 d/c ≤ τ (cid:48) ≤ d/c + (cid:15) . The implementation of Fig. 3 can be refined in thecase that the detectors at the left and right leaves belongto Alice and Bob, respectively, assuming that 2 n δ (cid:28) d . D D D D Bob’s knowledge on the bit-pair relations Bob’s information gainCase (1) A A B B x (cid:54) = y x = y x (cid:54) = y x = y x (cid:54) = y and x (cid:54) = y , or x = y and x = y x (cid:54) = y and x = y , or x = y and x (cid:54) = y x (cid:54) = y and x (cid:54) = y x , x ) (cid:54) = ( y , y ) 2 − log TABLE I. Some detector assignments and Bob’s corresponding information gains, given that one of Alice’s detectors clicks.In the cases (1) and (2) ((3) and (4)), Bob knows x ( x ) with certainty. In the cases (5) and (6), Bob knows x and x simultaneously with probability 0.5, which indicates that Bob can gain one-bit information on average. In case (7) Bob knowsthat x (cid:54) = y and x (cid:54) = y and his information gain is 2 (bits). In case (8), Bob knows that ( x , x ) (cid:54) = ( y , y ) and hence hisinformation gain is 2 − log δ , while the right paths have addi-tional delays denoted by longer optical fibers. That is, n cannot be arbitrarily large or n = O (log( d/δ )).Specifically, we can use only two detectors (one for Aliceand the other for Bob) and add a time domain coordinateto save the massive number of 2 n detectors required. Thisis done as shown in Fig. 5, where the left path at level i has a time delay δ and the right light path at level i has an additional delay of 2 n − i − δ and both the left andright paths the level n have delay δ . Consequently, the2 n light paths from left to right in Fig. 3 will have delays nδ, nδ, ( n + 1) δ, ( n + 1) δ, . . . , (2 n − − δ, (2 n − − δ inFig. 5, respectively. An important observation is thatat the same level each of Alice and Bob has the sameinput bit and applies the same PMs to the correspondinglight paths. Also a beam splitter has two input ports,which allows us to connect both branches to the samebeam splitter. Therefore, from the time of clicking, onecan deduct the corresponding light path in the circuit ofFig. 3 and learn the n -bit string.As a comparison, the previous scheme by Santo and Daki´c [19] uses one single-photon source and two detec-tors to exchange two bits of information. Our scheme isable to transmit more information at the cost of addi-tional fibers, MZ interferometers, and beam splitters. B. Information gains
Let us quantify how many detectors Bob shouldhave to optimize his information gain I ( X ; B | Y ), where I ( X ; B | Y ) = H ( B | Y ) − H ( B | X, Y ) is the mutual in-formation between Alice’s input variable X and Bob’soutput variable B conditioned on Bob’s input variable Y ; H ( B | Y ) is the conditional Shannon entropy , and H ( X ) = − (cid:80) x p x log p x . Let m be the number of detec-tors that belong to Bob. Since X and Y are independent,it is clear that I ( X ; B | Y ) ≤ H ( B ) = H / n , . . . , / n (cid:124) (cid:123)(cid:122) (cid:125) m , − m/ n = n − (cid:16) − m n (cid:17) log (2 n − m ) . The total information gain of Alice and Bob is I ( Y ; A | X ) + I ( X ; B | Y ) = H ( A ) + H ( B )= 2 n − m n log m − (cid:16) − m n (cid:17) log (2 n − m ) ≤ n + 1 , where the equality holds when m = 2 n − . The mainresult can be stated as follows: The optimal total information gain is n + 1 . To reach optimal total information gain, Alice and Bobeach should access half of the 2 n detectors. It does notmatter which detectors Alice or Bob should hold sincethe one with a clicking detector can learn n bits of in-formation, while the other learns one bit of information.As an illustration, we analyze Bob’s information gain inthe case of n = 2 as shown in Fig. 2. Various detectorassignments as listed in Table I, assuming that one ofAlice’s (Bob’s) detectors always clicks (never click).Note that, to win the GSD game with certainty, theone that cannot learn the other’s n input bits must knowone bit of information. With a delicate initial assignmentof these 2 n detectors between Alice and Bob, they canexchange the input bit pair ( x k , y k ) with certainty forsome specific k . Specifically, denote two detector sets by∆ Ek and ∆ Ok . For all i = 1 , . . . , n , the detector D i ∈ ∆ Ek ( D i ∈ ∆ Ok ) if D i receives a photon travelling through aneven-k (odd-k) path. Let all 2 n − detector elements in∆ Ek (∆ Ok ) are completely accessible to Alice (Bob). Inthis case, once none of the detectors belonging to Bobclicks. Bob can learn that that x k = y k and hence heoutputs the bit b k = x k with certainty. For example,as shown in Table I, Bob can always learn x using thedetector assignment in Case (1) or (2), or learn x usingthe detector assignment in Case (3) or (4).It is noteworthy to mention the following detector as-signment. Let Alice occupy only one detector and Boboccupy the other 2 n − − n , Al-ice’s detector receives a photon. In this case, the no-clickon Bob’s side makes him exclude the possibility of 2 n − x . Inother words, if Alice and Bob are tasked to output a = y and b = x , respectively, in the GSD game, they canwin the game with the probability 2 − n . On the otherhand, if one of Bob’s detectors clicks, Alice still learns n − log (2 n −
1) bits of information.
III. DISCUSSION
A lesson learned from the dense coding is that sendingone qubit is equivalent to sending two classical bits; an-other lesson from information causality is that, if thereis no quantum communication, the information gain isequal to the amount of classical communication. No-tably, the dense coding and random access code each (i)are one-way communication, and (ii) exploit quantumentanglement as physical resource. To the best of ourknowledge, the protocols of SD and GSD games are thefirst ones for two-way signaling quantum communication.Therein, the spatial coherent superposition and wave-particle duality can be regarded as physical resources.From the two-way signaling aspect, these two quantumproperties of a photon are more beneficial than quantumentanglement. In the proposed two-way signaling proto-col, sending a photon with an n -level circuit is equivalentto sending n + 1 bits, where n can be arbitrarily large.Which agent can obtain the others information dependson the local bit strings x and y , and the pre-assignmentof these 2 n detectors to Alice or Bob. In any way, thereis always a detector that clicks, which indicates either I ( Y ; A | X ) = n or I ( X ; B | Y ) = n must hold, and hencewe can conclude that n ≤ I ( Y ; A | X )+ I ( X ; B | Y ) ≤ n +1.From the causal perspective, the optimal informationgain in the GSD game can be explained in a two-foldway. Firstly, an information carrier is consumed therein.Notably, regarding the classical communication, infor- mation causality states that the information gain can-not exceed the amount of classical communication. Thussending-and-receiving a photon can result in one-bit in-formation gain. Secondly, the two distant local opera-tions at the same level fully determines into which waythe photon enters in the next level, and this contributesthe one bit information. In other words, only when thecoherent superposed parts meet at BS2 of a MZ inter-ferometer in every level as shown in Fig. 3, the which-way uncertainty between these two beam splitters in theMZ interferometer vanishes, and consequently producesone-bit information. That is, a level contributes one-bitinformation gain. At the end, at most ( n + 1) bits ofinformation can be generated during a photon enteringan n -level circuit.For example, in the Elitzur-Vaidman bomb tester, asingle photon is emitted, but one of its coherent parts isblocked and there is no interference at the second beamsplitter of a MZ interferometer [23]. In this case, onlya bit of information (whether the bomb explodes) is ac-cessible. On the other hand, in the simple one-way SDgame, assume that the bit y = 1 is public, and the bit x is unknown to Bob [19]. To inform Bob, Alice performslocal operations on the accessible coherent superposedpart. It is the interference at the second beam splitterbrings Bob the bit value of x . IV. IMPLEMENTATION
Since the complexity of the n -level circuit grows ex-ponentially in n (or linearly in n if the scheme of Fig. 5is used), it is impossible to realize the optical circuit forarbitrarily large n with imperfect devices. Noisy com-ponents, such as the photon source, beam splitters anddetectors, will cause the photon to decay and hence limitthe possible circuit level. FIG. 6. The rate of success of our GSD protocol versus thenumber of circuit levels.
Here we estimate the performance of the protocol whenit is implemented under realistic experimental condi-tions. We consider the following error sources. A realisticpulsed single-photon source has a photon number prob-ability P ( n ) to generate n photons per pulse. A quan-tum dot single-photon source can achieve P (1) = 0 . π/ (cid:15) per stageto be 1 . .
5% loss. We assume the detection effi-ciency η D of the detectors to be 85%, which is achievableusing superconducting nanowire single photon detectors(SNSPD). The contribution from the dark counts of thedetectors can be negligible by using low-dark-count de-tectors such as SNSPD or by applying gating techniques.Using these numbers, we obtain the success rate of ourprotocol for n stages to be P (1) (1 − (cid:15) ) n η D , as shown inFig. 6. Acknowledgments. — CYL was supported from theYoung Scholar Fellowship Program by Ministry ofScience and Technology (MOST) in Taiwan, underGrant MOST108-2636-E-009-004. YCC was supportedby MOST108-2218-E-009-035-MY3. [1] C. E. Shannon, Bell System Technical Journal , 379(1948).[2] R. Landauer, Phys. Today , 23 (1991).[3] L. Hardy, (2001), arXiv:quant-ph/0101012.[4] B. Daki´c and ˇC. Brukner, Deep Beauty: Understandingthe Quantum World through Mathematical Innovation ,365 (2011), arXiv:0911.0695.[5] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Phys.Rev. A , 012311 (2011).[6] L. Masanes, M. P. M¨uller, R. Augusiak,and D. P´erez-Garc´ıa, Proceedings of the Na-tional Academy of Sciences , 022119 (2017).[8] L.-L. Sun, X. Zhou, and S. Yu, (2019),arXiv:1906.11807.[9] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. ,2881 (1992).[10] A. S. Holevo, Probl. Peredachi Inf. , 3 (1973), Englishtranslation Problems Inform. Transmission , vol. 9, no. 3,pp.177–183, 1973.[11] R. Cleve, W. van Dam, M. Nielsen, and A. Tapp,in
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