Information Carrier and Resource Optimization of Counterfactual Quantum Communication
aa r X i v : . [ qu a n t - ph ] F e b Noname manuscript No. (will be inserted by the editor)
Information Carrier and Resource Optimization ofCounterfactual Quantum Communication
Fakhar Zaman · Kyesan Lee · HyundongShin
Received: date / Accepted: date
Abstract
Counterfactual quantum communication is unique in its own waythat allows remote parties to transfer information without sending any mes-sage carrier in the channel. Although no message carrier travels in the channelat the time of successful information transmission, it is impossible to transmitinformation faster than the speed of light, thus without an information carrier.In this paper, we address an important question “What carries the informa-tion in counterfactual quantum communication?” and optimize the resourceefficiency of the counterfactual quantum communication in terms of the num-ber of channels used, time consumed to transmit 1-bit classical informationbetween two remote parties, and the number of qubits required to accomplishthe counterfactual quantum communication.
Keywords
Counterfactual Quantum Communication · Resource Optimiza-tion · Interaction Free Measurement
Fakhar Zamanfakhar − In conventional communication protocols [1–4], information is carried by amessage carrier propagating from sender to receiver. Unlike conventional com-munication, counterfactual quantum communication enables the sender (re-ceiver) to transmit (receive) information without sending any message carrierin the channel [5–9]. Elitzur and Vadmin first introduced a counterfactualphenomenon in quantum mechanics [10] by investigating either the absorp-tive object is present or not in the Mach-Zehnder interferometer— interactionfree measurement (IFM). This invention enables one to interrogate the exis-tence of a bomb without interrogating it with a probability 1 /
2. Later, Kwiat et al. [11] increased this probability arbitrarily close to one by cascading anarray of N beam splitters in series to stabilize an unstable quantum stateby performing the frequent measurements— quantum Zeno effect [12, 13]. Theprotocol to transmit classical information in a counterfactual manner is basedon a nested version of the Mach-Zehnder interferometer with M outer and N inner cycles— chained quantum Zeno effect —and control the existence of anabsorptive object in the interferometer in classical manner [5, 14].In general, the number of channel usage η and the time T required to ac-complish the communication task are the key parameters to determine the effi-ciency of a communication system. In counterfactual quantum communicationprotocols, although the probability that the protocol is discarded approachesto zero as M and N increase, the performance parameters η and T increaseas follows η ∝ M N, (1) T ∝ ηT c , (2)where T c is the time for one round trip between the sender (Alice) and receiver(Bob). In this paper, we optimize the resource efficiency of counterfactualquantum communication. We determine the optimal values of M and N tominimize η and T for the given source probability q where q = P { b = 0 } and b is 1-bit classical information Alice wants to send to Bob. In counterfactualquantum communication, although no message carrier travels in the quantumchannel at the time of successful information transmission, the protocol needs aquantum channel to connect Alice to Bob. One may ask the following questionshere – If nothing is traveling in the quantum channel at the time of successfulinformation transmission, why do we need a quantum channel between theAlice and Bob? – What carries the information in counterfactual quantum communication?To answer the first question, we argue that there is a nonzero probability ǫ thatthe message carrier is found in the transmission channel where ǫ tends to zeroas M and N go to infinity. In case any message carrier is found in the quan-tum channel, the protocol discards. In response to the second question, herewe simply say the information is carried by the frequency of measurements itle Suppressed Due to Excessive Length 3 performed in the nested interferometer. The main contribution of this paperis to address the second question and show that even if no physical particleis traveled in the transmission channel, the information carrier is transmittedfrom Alice to Bob followed by the resource optimization of the counterfactualquantum communication. Currently, if the counterfactual quantum communi-cation protocol discards in the absence of the absorptive object, the classicalannouncement is the only way Alice comes to know the erasure event. At theend of the paper, we demonstrate that there is no need for a classical an-nouncement to notify Alice by making a slight modification in the originalprotocol and demonstrate its importance in the resource optimization of thecounterfactual quantum communication. The rest of the paper is organizedas follows. In Sec. 2, we demonstrate how the measurement frequency acts asan information carrier in counterfactual quantum communication followed byresource optimization of counterfactual quantum communication in Sec. 3. Inthe end, we summarize our results and discuss the future works. Counterfactual quantum communication is a unique phenomenon in quantummechanics that allows Alice and Bob to transfer information without trans-mitting any physical particle over the channel. To achieve the counterfactualquantum communication, Alice maps one bit of classical information b in thepresence or absence of the shutter in one arm of the interferometer. If b = 1,Alice introduces the shutter in her part of the interferometer to block the pathof the photon as shown in Fig 2. We consider that if the photon interacts withthe shutter, it absorbs the photon. In case b = 0, Alice allows the photon topass and reflect the component of the photon to Bob. Before going into thedetails of counterfactual quantum communication, we explain the working ofan interferometer and show that it is semi-counterfactual (counterfactual for b = 1 only). Fig. 1 shows an array of N unbalanced beam splitters (BS) andeach BS transforms the input state of the photon as U = (cid:20) cos θ N − sin θ N sin θ N cos θ N (cid:21) , (3)where θ N = π/ (2 N ). Consider the initial state of the photon is | ψ i = | i . Af-ter BS , the state of the photon is transformed as | ψ i = U | i = cos θ N | i +sin θ N | i . If b = 1, Alice blocks the path of the photon with shutter. Thepresence of the shutter is similar to measure the state of the photon in com-putational basis. If the state of the photon is | i , the photon is absorbed bythe shutter and no photon reaches at the second beam splitter. Unless thephoton is absorbed by the shutter, the photon collapses back to the initialstate and Bob inputs the photon to BS (see Fig. 1(a)). After N cycles, thephoton collapses to the initial state with probability cos N θ N which tends toone as N → ∞ and D clicks. Here it is important to note that Bob determines Fakhar Zaman et al.
Alice | i BS | i BS BS N D D ChannelBob (a) b = 1 Alice | i BS | i BS BS N D D ChannelBob (b) b = 0 Fig. 1
Semi-counterfactual quantum communication based on the quantum Zeno effect. Itis coutnerfactual only for b = 1. Here BS stands for unbalanced beam splitters. the presence of the shutter at Alice’s side without transmitting any physicalparticle over the channel.For b = 0, Alice allows the photon to pass and no measurement is per-formed on the photon. The N BSs transform the state of the photon to | ψ i = U N | i = | i and D clicks (see Fig. 1(b)). Bob decodes the classi-cal bit as b if D b clicks and the measurement frequency f is given as f = ( , for b = 0 ,N, for b = 1 . (4)However, this protocol is counterfactual for b = 1 only—called semi-counterfactualquantum communication . In case b = 0, the photon is found in the transmis-sion channel with certainty.To transfer both classical bits 0 and 1 in counterfactual way—called coun-terfactual quantum communication , consider the nested version of the interfer-ometer as shown in Fig. 2. Here white rectangles show the unbalanced beamsplitters BS O for outer cycles and gray rectangles denote the unbalanced beamsplitters BS I for inner cycles where each outer cycle has an array of N BS I .Each BS O and BS I transform the state of the photon as U O and U I , respec-tively where U O = cos θ M − sin θ M θ M cos θ M
00 1 , (5) U I = θ N − sin θ N θ N cos θ N , (6) itle Suppressed Due to Excessive Length 5 | i| i BS O Alice | i BS I BS I BS I N BS O BS O M BS I BS I BS I N D D ChannelBob (a) b = 1 | i| i BS O Alice | i BS I BS I BS I N BS O BS O M BS I BS I BS I N D D ChannelBob (b) b = 0 Fig. 2
Counterfactual quantum communication based on the chained quantum Zeno effect.It is counterfactual for both classical bits b = 0 and b = 1. Here white rectangles denote M outer cycles and gray rectangles represent N inner cycles. where θ M = π/ (2 M ). Assume that the quantum state of Bob’s photon can bein the superposition of | i , | i and | i where | i , | i and | i show the pathsof the photon (see Fig. 2). We consider that the initial state of the photon is | φ i = | i and Bob inputs the photon to the first outer cycle. After BS I ofthe first outer cycle, the initial state | φ i transforms as follows: | φ i = U I U O | i (7)= cos θ M | i + sin θ M cos θ N | i + sin θ M sin θ N | i . (8)If b = 1, Alice blocks the path of the photon with shutter (see Fig. 2(a)). Thepresence of the shutter is similar to perform the measurement on the photon.If the state of the photon is | i , the photon is absorbed by the shutter andno photon reaches at BS I . Unless the photon is absorbed by the shutter, thestate of the photon collapses to | φ i = cos θ M | i + sin θ M | i and the stateof the photon after BS I N of the first outer cycle collapses to | φ i . Bob inputsthe photon to BS O . After M outer cycles, the state of the photon is | φ i = U M − | φ i (9)= | i , (10)and detector D clicks. Note that the measurement frequency is f = N in eachouter cycle for b = 1.For b = 0, Alice allows the photon to pass and no measurement is per-formed on the photon. After BS I N of the first outer cycle, the state of the We consider cos θ N ≈ N . Fakhar Zaman et al. . . . . . . · − N ∆ [ b it s / c h a nn e l u s a g e ] , T / T c ∆ T/T c Fig. 3
Successful transmission rate ∆ [bits/channel usage] and
T/T c for q = 1 / M ⋆ =arg max M ∆. As N increases, the successful transmission rate ∆ decreases as a logarithmicfunction and T/T c increases linearly. photon transforms to | φ i = U N I U O | φ i (11)= cos θ M | i + sin θ N | i . (12)Alice blocks the path of the photon independent of the value of b (see Fig. 2(b)).Unless the photon is absorbed by the shutter, the state of the photon collapsesto the initial state | φ i . After M outer cycles, the state of the photon remainsin the initial state, and D clicks. Note that the measurement frequency is f = 1 in each outer cycle b = 0. At the end of the protocol, the measurementfrequency f is given as f = ( M, for b = 0 ,M N, for b = 1 . (13)Since the measurement on the quantum state disturbs the state of the systemunder observation, from (7) and (13), we can see that the measurement fre-quency on the qutrit itself is the information carrier in counterfactual quantumcommunication and transforms the initial state to orthonormal states whichare perfectly distinguishable to decode the information. In the previous section, we demonstrated that the measurement frequencycarries the information in the counterfactual quantum communication (see(13)). To ensure the full counterfactuality of the protocol, Alice and Bob usethe nested interferometer with M outer and N inner cycles (see Fig. 2) to itle Suppressed Due to Excessive Length 7 M Fig. 4 ζ M,N for q = 1 / P = 0 . M and N , there exist the optimal values of M and N such that { M ⋆ , N ⋆ } = arg min ζ M,N .For q = 1 / M ⋆ = N ⋆ = 2 minimizes ζ M,N as ζ min = 68. transfer one bit of classical information without transmitting any physicalparticle over the channel with an average success probability λ M,N for a givensource q where λ M,N = qλ + (1 − q ) λ , (14)and λ and λ are defined as functions of M and N as [7] λ = cos M θ M , (15) λ = M Y m =1 (cid:2) − sin ( mθ M ) sin θ N (cid:3) N . (16)Although no physical particle is found in the transmission channel at the timeof successful transmission, there is a nonzero probability that the photon hasone round trip between Alice and Bob in each inner cycle of the nested inter-ferometer which tends to zero as M and N go to infinity. Here we considerthe number of channel usages η and time T required to finish the task as theprimary resource of counterfactual quantum communication. Our goal is tominimize the number of channel usage and reduce the implementation timeto transfer classical information counterfactually. For a given source q , theresource parameters η and T to transmit 1-bit classical information counter-factually with success probability λ M,N are given as η = 2 M N, (17) T = 12 ηT c . (18) Fakhar Zaman et al. .
00 0 .
25 0 .
50 0 .
75 1 . q x m i n Fig. 5
Number of transmission trials x M ⋆ ,N ⋆ as a function of q for P = 0 . M and N . To determine the efficiency of the counterfactual quantum communication forgiven q corresponding to M and N , we define the successful transmission rate∆ [bits/channel usage] as ∆ = λ M,N η . (19)Fig. 3 shows the successful transmission rate ∆ [bits/channel usage] and
T /T c as a function of N corresponding to M ⋆ = arg max M ∆ = 2 and q = 1 / λ M,N increases with N , we can clearly see that ∆ decreases as alogarithmic function of N , on the other hand T /T c increases linearly.To optimize the resource efficiency of the counterfactual quantum commu-nication protocols, we fix the success probability P of counterfactual quantumcommunication independent of M and N . Let X be the number of transmissiontrials up to the success of counterfactual quantum communication. Then X isthe geometric random variable with success trial probability λ M,N . Let x M,N be the minimum required number of transmission trials for given P with M outer and N inner cycles. Then we have P P { X x M,N } (20)= x M,N X i =1 (1 − λ M,N ) i − λ M,N (21)= 1 − (1 − λ M,N ) x M,N , (22)leading to x M,N = & log (1 − P )log (1 − λ M,N ) ' . (23) itle Suppressed Due to Excessive Length 9 . . . ( N − T c T c MNT c | i AO | i AO b = 0 : | i AO | i AO b = 1 : Fig. 6
Wave functions for binary counterfactual quantum communication. Here | i AO denotes absence (presence) of the absorptive object and AO stands for the absorptive object. where ⌈·⌉ is the ceiling function. Since the success probability P is independentof M and N , we calculate ζ min = min ζ M,N , (24)where ζ M,N = M N x
M,N and the performance parameters can be optimizedas η min = 2 ζ min , (25) T min = ζ min T c , (26)∆ max = Pη min . (27)For example, ζ min = 68 for q = 1 / P = 0 .
975 with M ⋆ = N ⋆ = 2and x M ⋆ N ⋆ = 17 as shown in Fig. 4 where { M ⋆ , N ⋆ } = arg min ζ M,N . Here itis important to note that smaller the values of M ⋆ and N ⋆ , larger the valueof x M ⋆ ,N ⋆ , which reveals that a large number of photons needs to transfer 1-bit classical information counterfactually with success probability at least P .Fig. 5 shows the cost of minimizing η and T .In the original protocol for counterfactual quantum communication [5], ifthe photon is traveled between Alice and Bob for b = 1, it is absorbed by theabsorptive object (shutter) and the absorptive object jumps to high energylevel. It causes an erasure for both Alice and Bob. In case b = 0, if the photonis found in the transmission channel, it ends up at detector D (see Fig. 1in [5]). It causes an erasure for Bob only. Alice has no information either theinformation is transferred or the protocol is discarded. To perform the resourceoptimization, it is important that Alice knows when the first success occurs sothat she can start the transmission of the next bit in the string. To accomplishthis task, Fig. 6 shows two wave functions for binary counterfactual quantumcommunication. For b = 1, the wave function is the same as the presence ofthe absorptive object in the interferometer for M outer and N inner cycles. Incase b = 0, Alice transmits the wave function such that the absorptive object(shutter) is present in the last inner cycle of each outer cycle independent of b .With this wave function if the photon is found in the transmission channel for b = 0, it causes an erasure for both Alice and Bob. This slight modification at Alice’s side enables Alice and Bob to transmit one bit of classical informationunder the most optimal resource efficiency.
In this paper, we analyzed the resource efficiency of the counterfactual quan-tum communication and demonstrated that small values of M and N givebetter efficiency with multiple rounds of the counterfactual quantum commu-nication protocol ( x M,N ) to increase the success probability P . It is alreadywell known that the stability of the interferometer is questioned for large val-ues of M and N . In addition, the effect of channel noise [15,16] and photon lossprobability also increase as M and N increase. We showed that M = N = 2are the optimal values for q = 1 / P = 0 .
975 at the cost of x M,N = 17. Wealso demonstrated that the measurement frequency on the qutrit state acts asthe information carrier in the counterfactual quantum communication.
Acknowledgements
This work was supported by the National Research Foundation ofKorea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1A2C2007037).
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