Performance Evaluation of Quantum Merging: Negative Queue Length

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1 Performance Evaluaion of Quanum Merging: Negaive Queue Lengh Hiroshi Toyoizumi Waseda Universiy Nishi-waseda -6-, Shinjuku, Tokyo TEL: FAX: Absrac We sudy he deail of queues handling quanum communicaion, especially socalled quanum merging of qubis. In quanum merging, by using EPR pairs (enangled pairs of quanum paricles), jobs may bring exra service capaciy o he server, which can be sored in he sysem for fuure use. We will model his phenomena as queues ha can have he negaive queue lengh, and show he basic characerisics of hese quanum queues. Keywords: quanum merging, negaive enropy, negaive cusomer, queue, performance evaluaion, virual waiing ime Inroducion Quanum informaion processing is fundamenally differen wih is classical counerpar. The difference can be used o achieve somehing ha classical informaion processing canno achieve. For example, Shor s algorihm[8] use qubis (quanum version of bis) o solve facorizaion problem effecively, which is nooriously hard o be solved by classical compuers. This hardness is essenial o he safey of RSA algorihm[7], ha is used widely for secure communicaion on he inerne. On he oher hand, someimes quanum informaion processing shows us somehing srange o inerpre. One of hose examples, is enropy. In classical Shannon s heory, H(X), he enropy of a random variable X, is defined by H(X) x P {X x} log P {X x}. () The Shannon s enropy measures he uncerainy of a daa source, which is equivalen o how much informaion we can ge when we learn he value of X. Suppose we have some parial daa Y on our hand, which can be correlaed wih he source daa X. Then, given Y, he amoun of informaion we gained from X can be measured by he condiional enropy: H(X Y ) H(X, Y ) H(Y ). () For example, if H(X Y ) 0, hen X f(y ), which means we already have complee informaion of X. Thus, inuiively (and mahemaically provable), H(X Y ) is always nonnegaive. In he quanum world, he Neumann s enropy S(ρ) and condiional enropy S(ρ σ) are defined by S(ρ) r(ρ log ρ), (3) S(ρ σ) S(ρ, σ) S(σ), (4) for he densiy operaors of quanum saes ρ and σ. Alhough he Neumann s enropy is a naural exension of Shannon s enropy and measures he informaion carried, S(ρ) can be negaive for EPR pairs (p.54 in [6]). In oher words, we have he informaion more han cerain for EPR pairs, which has been he mysery of quanum informaion heory. In 005, Horodecki, Oppenheim and Andreas proposed he concep of parial informaion in [5]. They also inroduced he framework called quanum merging o solve he mysery of negaive quanum enropy. They showed ha in he case when S(ρ) < 0, hey can sore he excess of he cerainy in he bank and use hem in he fuure. Thus, he performance of quanum merging is cerainly differen wih he classical informaion processing. In his paper, we discuss he performance of he queue for quanum merging in deail, using queues ha can have he negaive queue lengh. In 99, Gelenbe inroduced he concep of negaive cusomers in Markovian nework seing[3]. There, a negaive cusomers are supposed o erase one posiive cusomer in queue while he queue lengh is posiive. The sabiliy and produc form soluion are sudied in he papers like [3,, ]. However, he negaive cusomers in our model can be kep in he queue for he fuure use, even when here is no

2 posiive cusomers. Tha is he main new concep derived from quanum merging. Quanum Communicaion by Qubi Merging We inroduce he framework of quanum merging proposed in [5]. Suppose Alice wans o ransfer he sae of qubis o Bob. Bu in his case, Bob has his qubi, which is probably correlaed wih he sae of Alice s. In oher words, Alice wans o send some informaion o Bob and merge her sae o Bob s. Le ρ A and ρ B be Alice and Bob s densiy operaor and le ρ AB be he join quanum sae of Alice and Bob. Alice can use boh he quanum channel and classical channel. Depending on he Bob s quanum sae, he procedure is differen. Le us consider he following hree cases. To disinguish he bis, we use A for he Alice s and B for Bob s bi.. Independen saes (S(ρ A ρ B ) ). In his case, Alice has a random bi: ρ A ( 0 0 A + A ), (5) and Bob has his some sae: The join densiy marix is ρ B 0 B. (6) ρ AB ( 0 0 A + A ) 0 B. (7) In his case, Alice has o send one qubi wih he quanum channel o merge her sae.. Classically srongly-correlaed saes (S(ρ A ρ B ) 0). In his case he join quanum sae is ρ AB ( AB + AB ). (8) The mixed sae can be considered a par of pure sae inroducing a reference sysem R. The sae is already synchronized. Thus, Alice doesn have o send any quanum informaion. Alice need o do some measuremen (maybe for a reference sysem), hen she can use classical channel o send he resul o Bob, hen Bob can adjus his sae locally o merge Alice s sae. 3. Enangled sae (S(ρ A ρ B ) ). The join quanum sae is a pure and enangled sae, for example, β 00 ( 00 + ). (9) Thus, i is clear ha in quanum merging, he arrival of jobs (sending qubis) is no necessary a workload, bu also enhance he service capaciy, which is radically differen from he classical bi merging. 3 Queues wih Negaive Queue Lengh We model he quanum merging scheme and evaluae is performance using queueing heory. 3. Queuing Model and Negaive Virual Waiing Time Suppose Alice and Bob build up a sysem for a quanum merging and open i for public. Cusomers arrive a he sysem carrying a qubi o merge o he receiver end. For he simpliciy, we assume wo ypes of cusomers:() whose qubi is independen wih he receiver end, () whose qubi is enangled wih he qubi a he receiver end. The firs is called posiive cusomers and he laer is called negaive cusomers. The boh posiive and negaive cusomers arrive a he sysem wih independen Poisson processes wih he rae λ p and λ n, respecively. We can assume anoher ype of cusomers whose qubi is classically srongly-correlaed. However, hose cusomers doesn have o use quanum communicaion, so we ignore here. As showed in he Secion, he posiive cusomer has o send his sae o he receiver wih a quanum channel, which is required some service ime S. Maybe his will be done by using a phoonic cable by sending a new EPR pair. The service ime S is assumed o be independen and idenically disribued wih G(x) P {S x}. On he oher hand, a negaive cusomer has a EPR pair, which can be used for sending he quanum sae of a posiive cusomer by using he quanum eleporaion. We assume whenever here is a posiive cusomer waiing, hose EPR pair of he negaive cusomers will be used. When here is no posiive cusomer in he sysem, we can save EPR pairs for fuure use. The queueing discipline is assumed o be firs-come-firs-serve. Le N() be he number of cusomers in he sysem. We allow N() o be negaive. When he queue lengh N() is negaive, we have a pool of EPR pairs or negaive cusomers in he sysem. The queue can be called an M/G/± queue for indicaing o have boh posiive and negaive queue lengh. Here is he summary of he sysem ransiions:. When a posiive cusomer sees N() < 0 a his arrival, hen he consumes one negaive cusomer o send his qubi and leaves he sysem immediaely. In his case, his EPR sae is no only synchronized, bu also has he abiliy o ransfer oher qubi (as a ool of quanum eleporaion), while Bob creaes a new EPR pair locally.. When a posiive cusomer sees N() 0, hen he joins he queue and wais for his service. His service will be erminaed by he normal service compleion required S, or by an arrival of negaive cusomer.

3 3. When a negaive cusomer sees N() > 0, hen he cusomer in service finishes he service immediaely and leave he sysem. 4. When a negaive cusomer sees N() 0, hen he will join he (negaive) queue. V n() posiive arrival negaive arrival V() posiive arrival negaive arrival posiive deparure Figure : V n (): Virual waiing ime in he negaively censored process. Figure : Virual waiing ime in a queue wih negaive queue lengh. Now we exend he definiion of he virual waiing ime V () for he queues wih negaive queue lengh. Suppose an addiional virual posiive cusomer has been arrived a our sysem a he ime. We allow his cusomer o arrive prior o he ime. The virual waiing ime V () is he difference beween and he ime when his virual cusomer sars his service. When he queue lengh is posiive, he sar ime of his service is laer han, so V () is he same as he ordinary virual waiing ime. However, when he queue lengh is negaive, his virual cusomer could have sared his service a he arrival of he negaive cusomer who arrived earlies in he queue. Thus, V () can be boh negaive and posiive. In Figure, we depic a sample pah of our virual waiing ime process. We will evaluae he mean of he saionary version of he virual waiing ime E[V ()] by censoring he virual waiing ime process. 3. Censoring and Idle-ime Modificaion In order o use ordinary queuing heory, we will break up our virual waiing ime process ino pieces. Le us exend he noion of busy period in he case of V () < 0. Thus, when he sysem is idle, boh negaive and posiive cusomers can sar a new busy period. Since he posiive and negaive arrivals are Poisson process, he sars of new posiive busy period are renewal poins. We will pick only busy cycles sared by negaive cusomers and pu hem ino one process (see Figure ). Le V n () be he virual waiing ime of his censored process. i is easy o see ha V n () is nohing bu he aained sojourn ime process. The aained sojourn ime is probabilisically equivalen wih he virual waiing ime, especially, he lengh of corresponding busy period lengh is he same [9]. Moreover, his negaively censored process has he same probabilisic srucure wih M/M/ queues excep he idle period. Noe ha idle periods are exponenially disribued wih he mean /(λ p + λ n ), no /λ n. Now we will modify he idle periods. We will ake ou all he idle period, hen subsiue hem wih new exponenial random variables wih he mean /λ n, which is independen wih ohers. Noe ha his modificaion will no affec he sample pah of virual waiing ime in he busy period. The resuled modified process is an M/M/ queue wih he arrival rae λ n and he service rae λ p. Le Y n be he busy period of his modified process, hen given λ p > λ n, we have E[Y n ] λ p λ n, (0) (for example see [0, 4]). Furher, le W n () be he saionary version of he virual waiing ime of he modified process, hen E[W n ()] λ n/λ p λ p λ n. () By adjusing he idle period replacemen, we can obain he mean virual waiing ime of negaive process, V n () as E[Y n ] + /λ n E[ V n ()] E[Y n ] + /(λ p + λ n ) E[W n()] λ n + λ p λ p (λ p λ n ). () Now we consider he posiive par. Jus like he negaive busy cycles, we will ake ou he posiive

4 cycles and pu hem ino one process. Le V p () be he he virual waiing ime of his posiive process. See Figure 3 V p() posiive arrival negaive arrival posiive deparure Figure 3: V p (): Virual waiing ime in he posiively censored process. Nex we modify all idle period wih new exponenial random variables wih he mean /λ p. The resuled modified process is an ordinary M/G/ queue wih he arrival rae λ p and he i.i.d. service ime S p defined by S p min(s, T n ), (3) where T n is he ime lengh o he nex negaive arrival and is he exponenial random variable wih is mean λ n. Le µ p /E[S p ], Y p be he lengh of busy period in his modified M/G/ queue. Then, E[Y p ] µ p λ p. (4) Le W p be he virual waiing ime of he modified M/G/ queue, hen given λ p < µ p, we have E[W p ] λ pe[s p]/ λ p /µ p, (5) by Pollaczek-Khinchine formula (see [0, 4]). Adjusing he replacemen of he idle ime, we have E[V p ()] E[Y p ] + /λ p E[Y p ] + /(λ p + λ n ) E[W p] (λ p + λ n )E[S p]/ ( + λ n /µ p )( + λ p /µ p ). (6) 3.3 Busy Cycle ha Arrivals See Le P p be he probabiliy ha a cusomer sees a posiive busy cycle, and P n be he probabiliy ha a cusomer sees a negaive one. A he sar of he busy period, a posiive cusomer sars busy period wih he probabiliy λ p /(λ p + λ n ). Since he arrival of cusomers is assumed o be a Poisson process, we have Also, P p (λ p λ n )(µ p + λ n ) (λ p + λ n )(µ p λ n ). (7) P n P p λ n (µ p + λ n ) (λ p + λ n )(µ p λ n ). (8) Noe ha hese probabiliies are insensiive wih he second momen of he service ime S p. I is easy o see ha he condiion λ p < µ p will guaranee P p <, since we have always µ p /E[min(S, T n )] > λ n. When he arrival rae of negaive cusomers ges bigger and λ n λ p, we have P p 0 and P n as expeced. On he oher hand, when he arrival rae of posiive cusomers approaches o he service rae µ p, we have P p and P n Virual Waiing Time By condiioning on he busy cycle ha a cusomer sees, we have E[V ()] E[V () an arrival sees a posiive cycle]p p + E[V () an arrival sees a negaive cycle]p n. (9) Given ha an arrival sees a posiive cycle, V () is equivalen o V p () in disribuion. Thus, by using (6), we have E[V () an arrival sees a posiive cycle] E[V p ()] (λ p + λ n )E[S p]/ ( + λ n /µ p )( + λ p /µ p ). (0) Similarly, by using () E[V () an arrival sees a negaive cycle] E[V n ()] λ n + λ p λ p (λ p λ n ). () Hence, we have he mean virual waiing ime of his sysem: E[V ()] µ p(λ p λ n )E[Sp]/ (µ p λ n )(µ p λ p ) λ n (µ p λ p ) λ p (µ p λ n )(λ p λ n ). () Noe ha when we have no negaive cusomers, λ n 0 and E[S p] E[S ] in (). Thus, we have E[V ()] λ pe[s ]/ λ p E[S]. (3) which is he usual Pollaczek-Khinchin formula for M/G/ queues.

5 Le W be he he ordinary waiing ime for cusomers in he saionary sae. Noe ha negaive cusomers always see no waiing ime. Since he arrival is Poisson, W max(v (), 0) in disribuion. Thus, λ p E[W ] E[V p ()]P p λ p + λ n λ p µ p(λ p λ n )E[Sp]/ (µ p λ n )(µ p λ p )(λ p + λ n ), (4) where λ n < λ p < µ p. 4 Example of an M/D/± queue E W Ρ Figure 4: Mean waiing ime in he M/D/± queue. The percenage shows he raio of posiive cusomers in he arrival cusomers. The 00% corresponds o he ordinary M/D/ queue. Le us consider an M/D/± queue. The service ime of cusomers S is consan, ha is, S /µ. In his case, /µ p E[S p ] and E[S p] in (4) can be calculaed by and E[S p ] E[min(S, T n )] λ n { e λn/µ }. (5) E[S p] λ n (µ + λ n)e λn/µ λ. (6) nµ 30% 70% 00% References [] C Benne, G Brassard, C Crepeau, R Jozsa, A Peres, and W Wooers. Teleporing an unknown quanum sae via dual classical and EPR channels. Phys Rev Le, pages , 993. [] P. Glynn E. Gelenbe and K. Sigmann. Queues wih negaive cusomers. Journal of Applied Probabiliy, 8:45 50, 99. [3] E. Gelenbe. Produc-form queueing neworks wih negaive and posiive cusomers. Journal of Applied Probabiliy, 8: , 99. [4] L. Kleinrock. Queueing Sysems Vol.. John Wiley and Sons, 975. [5] Jonahan Oppenheim Micha Horodecki and Andreas Winer. Parial quanum informaion. Naure, 436: , Augus 005. [6] Isaac L. Chuang Michael A. Nielsen. Quanum Compuaion and Quanum Informaion. Cambridge Univ Pr, 000. [7] R. L. Rives, A. Shamir, and L. Adleman. A mehod for obaining digial signaures and public-key cryposysems. Commun. ACM, 6():96 99, 983. [8] Peer W. Shor. Algorihms for quanum compuaion: Discree logarihms and facoring. In IEEE Symposium on Foundaions of Compuer Science, pages 4 34, 994. [9] Hiroshi Toyoizumi. Sengupa s invarian relaionship and is applicaion o waiing ime inference. J. Appl. Probab., 34(3): , 997. [0] R.W. Wolff. Sochasic modeling and he heory of queues. Princeon-Hall, 989. [] M. Miyazawa X. Chao and M. Pinedo. Queueing Neworks, Cusomers, Signals and Produc Form Soluions. John Wiley, 999. Figure 4 shows he mean waiing ime of M/D/± queues wih S. Unlike he ordinary M/D/ queue, he mean waiing ime will no blow up for ρ λ/µ >. We can see posiive effec of negaive cusomers here. 5 Conclusion We develop a queueing model for quanum merging. The queueing model we proposed is he M/G/± queue ha can have he negaive queue lengh and negaive virual waiing ime. We derive he explici form of he mean waiing ime of as well as he mean virual waiing ime in his M/G/± queue.