Strong supremacy of quantum systems as communication resource
SStrong supremacy of quantum systems as communication resource
Maria Quadeer and Manik Banik
Optics & Quantum Information Group, The Institute of Mathematical Sciences,HBNI, C.I.T Campus, Tharamani, Chennai
600 113 , India.
Andris Ambainis and Ashutosh Rai
Center for Quantum Computer Science, Faculty of Computing,University of Latvia, Raina bulv. , Riga, LV- , Latvia. We investigate the task of d -level random access codes ( d -RACs) and consider the possibility ofencoding classical strings of d -level symbols (dits) into a quantum system of dimension d (cid:48) strictly lessthan d . We show that the average success probability of recovering one (randomly chosen) dit fromthe encoded string can be larger than that obtained in the best classical protocol for the task. Ourresult is intriguing as we know from Holevo’s theorem (and more recently from Frenkel-Weiner’sresult [Commun. Math. Phys. , ( )]) that there exist communication scenarios whereinquantum resources prove to be of no advantage over classical resources. A distinguishing feature ofour protocol is that it establishes a stronger quantum advantage in contrast to the existing quantum d -RACs where d -level quantum systems are shown to be advantageous over their classical d -levelcounterparts. PACS numbers: . .Ud, . .Le, . .Ac I. INTRODUCTION
Information theory fundamentally deals with theproblem of reproducing at one point, either exactly orapproximately, a message selected at another point. Themathematical model for classical information theory wasfounded in a seminal paper by Claude Shannon in theyear 1948 [ ]. In the last few decades, application ofquantum theory to information processing tasks haslead to several discoveries [ – ] and their synthesis hasgiven birth to the epoch of quantum information theory[ , ]. Suitable use of resources from quantum theorycan outperform their classical counterparts in several in-formation theoretic, communication, and computationaltasks and some of these proposals have already beenimplemented by the present day quantum technologies[ , ].The advantage of quantum mechanical systems overclassical systems may be ascribed to the fact that thestate of a quantum system is given by a unit vector insome complex vector space. However, some informa-tion processing tasks in quantum theory are restricted.For example, the Holevo’s theorem [ ] puts a restric-tion on the amount of classical information that can beextracted from a quantum state ( accessible information ).More recently, a remarkable theorem due Frenkel andWeiner [ ] shows that the classical information stor-age in a d -Level quantum system can not be more thanthe corresponding d -state classical system. Despite ofsuch restrictions, a quantum system can give advantagein a suitably designed communication task known as Random Access Codes (RACs). These were initially in-troduced by Weisner by the name of conjugate coding [ ], and was later rediscovered in [ , ] by Ambainis et al. As pointed out in [ ], the possibility of encod-ing infinite amount of classical information in a singlequantum state (a vector in a complex Hilbert space) andthe freedom to perform different non-commutative meas-urements for extracting the encoded information renderquantum random access codes (QRACs) advantageous.QRACs establish that Bennett’s first law of quantum in-formation, i.e., “1 qubit (cid:23) ] (here X (cid:23) Y readsas “ X can do the job of Y ") is actually strict, i.e., as acommunication resource 1 qubit outperforms 1 bit inRAC tasks. Recently, Tavakoli et al. have studied RACswith high-level symbols that use d -level classical andquantum systems [ ]. It has been proved that high-level quantum systems provide significant advantage inthe average performances of the RAC tasks over theirclassical counterparts [ , ]. In other words, gener-alized version of Bennett’s first law, that “1 qudit (cid:23) d -level RAC quantum systemoutperforms its classical counterpart.Here, we ask whether a d (cid:48) -level quantum system canoutperform a d -level classical system in some commu-nication task, where d (cid:48) < d . Interestingly, we find thatthe answer is in the affirmative. We show that for high-level RACs there exist quantum codes that use relativelylower-level quantum systems for encoding to give bet-ter average success probability compared to the bestclassical codes that use high-level classical systems forencoding. This establishes an advantage of lower levelquantum systems as communication resources over ahigher level classical system. It is important to note thatthe quantum supremacy established by our protocolis stronger than that established in the existing QRAC a r X i v : . [ qu a n t - ph ] M a r protocols. In other words, we can say that in certaincommunication tasks 1 qu- d (cid:48) -dit (cid:31) d -dit with d (cid:48) < d .The paper is organized as follows: We briefly reviewRACs in section-II; in section-III, we discuss the high-level RACs; in section-IV, we present our protocol for im-plementing high-level RACs with lower-level quantumsystems, and finally give our conclusions in section-V. II. RANDOM ACCESS CODES: A QUICK OVERVIEW
Random access codes (RACs) are a class of communic-ation tasks involving two separated parties, (say) Aliceand Bob. Alice is given an n -bit string x = x ... x n chosenuniformly at random from the set {
0, 1 } n . The otherparty, Bob is given a number y ∈ {
1, ..., n } , chosen uni-formly at random. Bob’s task is to correctly guess the y th bit of Alice. Alice can help Bob in guessing the bitby sending some information about her string. However,the amount of information that Alice can send to Bobis restricted to 1-cbit. We denote such a RAC by thesymbol [ n p −→ ] , meaning that n bits are encoded into 1bit and p denotes the merit of the success of recoveringinitial bits which can be either ‘average success prob-ability’ ( P a ) or ‘worst case success probability’ ( P w ). Inthe quantum version of RAC, Alice can encode her n -bitstring into a two level quantum system, i.e., a qubit. Onecan also define a more general version of RAC denotedby [ n p −→ m ] (with m ≤ n ), which is defined analogousto the [ n p −→ ] RACs [ , , ]. Throughout this paperwe consider RACs with m = p −→ P Cw = P Ca = C (cid:48) to denote the classicalcase. If Alice sends one of the bit in her string (eitherfirst or second bit), Bob can guess that bit correctly buthe has to guess the other bit randomly. This naive pro-tocol gives the aforementioned optimal classical successprobability (both worst and average), and moreover onecan show that no other classical protocol can do better[ ]. Interestingly, the authors in [ , ] have shownthat for such a classical RAC non-trivial quantum pro-tocols exist which is described as the following : Alicecan encode her string x x into the state | ψ (cid:105) ∈ C of atwo-level quantum system using the following encodingscheme : x x → | ψ x x (cid:105) : = σ x X σ x Z | ψ (cid:105) ,where σ X , σ Z are Pauli operators and | ψ (cid:105) : = ( | (cid:105) + | X (cid:105) ) / (cid:112) + √
2, with | (cid:105) ( | X (cid:105) ) being the ‘up’ eigen-state of Pauli- Z (Pauli- X ) operator. To decode the firstbit, Bob, after receiving the encoded particle from Alice,performs a σ Z measurement (i.e performs measurement in computational basis {| (cid:105) , | (cid:105)} ) on it and guesses thebit value to be 0 ( ) upon obtaining the ‘up’ (‘down’)outcome. For the second bit, Bob performs σ X meas-urement and similarily guesses the outcome. For thisencoding-decoding scheme, the worst case success prob-ability is P Qw = (cid:16) + √ (cid:17) > = P Cw . It is easy to seethat the average success probability is the same as thatof the worst case, thus P Qa = (cid:16) + √ (cid:17) > P Ca . Later,Chuang (as mentioned in [ ]) generalized this protocolto a 3 −−→ et al. [ ] proved thatno quantum n p −→ n ≥ p w > n p −→ n ≥ p w > ].At this point we would like to mention that QRACshave been originally studied in the context of quantumfinite automata [ , ]. They also have applications inquantum communication complexity [ – ], in particu-lar for network coding [ ] and locally decodable codes[ , ]. QRACs have also been applied to the quantumstate learning problem [ ]. In recent times, this studyalso finds applications in semi-device-independent ran-dom number expansion [ ] as well as semi-device-independent key distribution [ ]. Also, the study ofQRACs and it’s variant, namely parity oblivious RACshas foundational implications [ – ], in particular theyhave been studied for operational depiction of prepara-tion contextuality of mixed quantum states. III. HIGH-LEVEL RANDOM ACCESS CODES
Recently Tavakoli et al. have studied high-level RACsin Ref.[ ]. Alice receives an n -dit string x = x , ..., x n ,uniformly at random, where a dit i.e x i takes valuesfrom an alphabet set {
0, 1, ..., d − } . She then encodesher string x into a classical d -level system (or a quantum d -level system), which she can send to Bob in any stateshe wishes. Bob receives a number y ∈ {
1, ..., n } chosenuniformly at random and his task is to recover the y th dit, i.e., x y of Alice’s string. Such a task is denotedby [( n , d ) → ] RAC. As conjectured in [ ] and laterproved in [ ], the maximum average success probabilityfor classical RACs is achieved by the ‘majority-encodingidentity-decoding’ protocol. A closed analytical formulafor the classical average success probability in this taskis hard to derive for general values of parameters n and d . However, for [( d ) → ] and [( d ) → ] cases, theanalytic expressions can be obtained for the maximumsuccess probabilities, that read P Ca ( d ) = ( + d ) and P Ca ( d ) = ( + d − d ) respectively. The authors in[ ] have also constructed non-trivial quantum pro-tocols for [( d ) → ] and [( d ) → ] cases. The [( d ) → ] quantum protocol is a generalized version ofthe [(
2, 2 ) → ] protocol already discussed in the previ-ous section. We describe the protocol here for complete-ness: Consider the computational basis B C : = {| l (cid:105)} d − l = ,in the Hilbert space C d and also consider the Fourierbasis B F : = {| e l (cid:105) = √ d ∑ d − k = ω kl | l (cid:105)} d − l = , with ω beingthe d th root of unity, i.e., ω = exp ( π id ) . Consider theoperators X = ∑ d − k = | k + (cid:105)(cid:104) k | and Z = ∑ d − k = ω k | k (cid:105)(cid:104) k | .Alice encodes her strings in the following manner: x x → | ψ x x (cid:105) = X x Z x | ψ (cid:105) , ( )where | ψ (cid:105) = N d ( | (cid:105) + | e (cid:105) ) , with N d = (cid:112) + √ d being the normalization constant. For decoding the firstdit x Bob performs a measurement in the computa-tional basis B C and guesses the value as l when theprojector | l (cid:105)(cid:104) l | clicks in the measurement. Similarly, forthe second dit, he performs a measurement in the Four-ier basis B F and guess the dit value according to themeasurement outcome. This protocol gives average suc-cess probability P Qa ( d ) = ( + √ d ) which is strictlygreater than the corresponding optimal classical successprobability P Ca ( d ) = ( + d ) for all d . As pointed outin [ ], the optimal advantage of the QRAC over theRAC measured by the ratio of the success probabilitiesis observed for d = IV. HIGH-LEVEL RAC WITH LOWER-LEVELQUANTUM ENCODING
We consider the [( d ) → ] RAC task, but Alice hasaccess to a d (cid:48) -dimensional quantum system to encodeher classical message where the dimension d (cid:48) of thequantum system is strictly less than d . We investigatewhether with a limited dimensional quantum systemAlice and Bob can construct non-trivial quantum codesfor the task. It turns out that the following QRAC pro-tocol has this interesting feature. Alice’s encoding : Alice encodes her string x x ∈{
0, 1, ..., d − } as follows: x x → | ψ x x (cid:105) : = G ( x , x , X , Z ) | ψ (cid:105) ∈ C d (cid:48) ( )with | ψ (cid:105) being the normalized state considered inEq.( ), and G ( x , x , X , Z ) = (cid:40) X x Z x , if both x , x ≤ d (cid:48) , otherwise.with X , Z being the operators as defined in the previoussection, and is the identity operator. Bob’s decoding : To decode the first dit Bob performsmeasurement in the computational basis {| l (cid:105)} d (cid:48) − l = andon obtaining outcome l ∈ {
1, ..., d (cid:48) − } he guesses the d ′ r = d - d ′
01 0 10 20 30 40 50 dd Figure . (Color online) : Plot r vs d . The plot (red dots) givesthe dimensional advantage r = d − d (cid:48) of the quantum systemused for encoding as a function of d . The minimum value of d for which a restricted quantum encoding is advantageousover the best classical d -RAC is d =
6. This advantage ingeneral increases with increasing d although remains flat insome range of d . For example, r = d ∈ {
6, ..., 11 } andthen increases to r = d ∈ {
12, ..., 19 } , etc. value as l , but when he obtains the outcome l = { d (cid:48) , d (cid:48) +
1, ..., d − } uniformly at random. To guess the value of thesecond dit, he performs a measurement in the Fourierbasis {| e l (cid:105) = √ d ∑ d (cid:48) − k = ω kl | l (cid:105)} d (cid:48) − l = and uses the samestrategy as before to make a guess. The average successprobability of this quantum protocol turns out to be: P Qres = d − r d (cid:20) + √ d − r (cid:21) , ( )where we call r = d − d (cid:48) a dimensional advantage sinceit gives the extent to which one can lower the dimen-sion of the quantum system used for encoding and stillget advantage over the best classical protocol. For anyclassical [( d ) → ] RAC the optimal average successprobability is P Ca ( d ) = ( + d ) . Therefore, in orderto do better, our quantum protocol must satisfy P Qres > P Ca ( d ) = ⇒ d > r + r +
1. ( )Condition-( ) implies that our quantum pro-tocol gives advantage for d ≥ r ∈ (cid:110)
1, ..., (cid:98) ( − + √ d + ) (cid:99) (cid:111) , where (cid:98) a (cid:99) is the greatestinteger less then or equal to a . This means that thesmallest value of d (cid:48) that yields an advantage is restricted.For small values of d , the allowed value of d (cid:48) is exactly d −
1, as dictated by condition-( ) (See Fig. ). As d increases, the allowed values of d (cid:48) become d − d − any lower d (cid:48) dimensional quantum system to supersedethe optimal classical d -RAC protocol. In fact, thereexists a minimum value of d (cid:48) , encoding below whichwould be bad, since it would yield a success probabilitylower than that of the optimal classical d -RAC. Fig. illustrates how the dimensional advantage r = d − d (cid:48) grows with d . Note that for d = {
2, 3, 4, 5 } we get nodimensional advantage (In Fig. , r = V. CONCLUDING REMARKS
In conclusion, our protocol demonstrates the suprem-acy of a lower-dimensional quantum system over it’shigher-dimensional classical counterpart, a consequenceof the existence of superposition of states and non-commutative measurements in quantum theory. Ourresult is slightly stronger than the earlier results onRACs which prove the advantage of using a quantumsystem over a classical system of the same dimension[ ]. In general, the study of QRACs is important asthere exist communication scenarios where the use ofquantum resources proves to be of no advantage over their classical counterparts, a result of Holevo’s theorem[ ], a general version of which has been proved morerecently by Frenkel and Weiner [ ]. While these the-orems address the question of encoding and decodinga random variable, RACs are concerned with decodingrandomly chosen parts of such an encoded random vari-able. Finally, we would like to state that proving theoptimality of high-level RACs using a limited dimen-sional quantum system remains an open problem andneeds further investigation. ACKNOWLEDGMENTS
MB and AR would like to thank Prof. G. Kar for hiscomments and suggestions. MB acknowledges his visitat the University of Latvia where a part of the workwas done. MQ acknowledges her visit at the Institute ofMathematical Sciences, Chennai. AR and AA acknow-ledge support by the European Union Seventh Frame-work Programme (FP / - ) under the RAQUEL(Grant Agreement No. ) project, QALGO (GrantAgreement No. ) project, and the ERC AdvancedGrant MQC. [ ] C. E. Shannon, A Mathematical Theory of Communic-ation, The Bell System Technical Journal , – ; – ( ).[ ] C. H. Bennett and S. J. Wiesner, Communication via one-and two-particle operators on Einstein-Podolsky-Rosenstates, Phys. Rev. Lett. , ( ).[ ] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres,and W. K. Wootters, Teleporting an Unknown QuantumState via Dual Classical and Einstein–Podolsky–RosenChannels, Phys. Rev. Lett. , ( ).[ ] L. Grover, A fast quantum mechanical algorithm for data-base search, In Proceeding of STOC’ , pp – (quant-ph/ ).[ ] P. W. Shor, Polynomial-Time Algorithms for Prime Factor-ization and Discrete Logarithms on a Quantum Computer,SIAM J. Sci. Statist. Comput. , ( ).[ ] A. Ambainis, Quantum Walks and their Algorithmic Ap-plications, Int. J. Quantum Inform. , ( ).[ ] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information, Cambridge University Press( ).[ ] M. M. Wilde, Quantum Information Theory, CambridgeUniversity Press ( ).[ ] M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M.Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri,and D. J. Wineland, Deterministic quantum teleportationof atomic qubits, Nature , ( ).[ ] J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, Beating thechannel capacity limit for linear photonic superdensecoding, Nature Physics , ( ). [ ] A. S. Holevo, Bounds for the quantity of information trans-mitted by a quantum communication channel, Problemsof Information Transmission , ( ).[ ] P. E. Frenkel and M. Weiner, Classical Information Storagein an n-Level Quantum System, Commun. Math. Phys. , ( ).[ ] S. Wiesner, Conjugate coding, SIGACT News , ( ).[ ] A. Ambainis, A. Nayak, A. Ta-Shama, U. Vazirani, DenseQuantum Coding and a Lower Bound for -Way QuantumAutomata, Proceedings of st ACM Symposium on Theoryof Computing , pp. - , .[ ] A. Ambainis, A. Nayak, A. Ta-Shama, and U. Vazirani,Dense Quantum Coding and Quantum Finite Automata,Journal of the ACM , ( ).[ ] A. Tavakoli, A. Hameedi, B. Marques, and M. Bourennane,Quantum Random Access Codes Using Single d -LevelSystems, Phys. Rev. Lett. , ( ).[ ] A. Ambainis, D. Kravchenko, and A. Rai, Optimal Clas-sical Random Access Codes Using Single d-level Systems,arXiv: . .[ ] A. Nayak, Optimal Lower Bounds for Quantum Automataand Random Access Codes, Proceedings of the th IEEESymposium on Foundations of Computer Science (FOCS’ ) ,pp. - , (e-print).[ ] M. Hayashi, K. Iwama, H. Nishimura, R. Raymond, andS. Yamashita, ( , )-Quantum random access coding does not exist—one qubit is not enough to recover one of fourbits, New J. Phys. , ( ).[ ] A. Ambainis, D. Leung, L. Mancinska, and M. Ozols,Quantum Random Access Codes with Shared Random-ness, arXiv: . .[ ] E. F. Galvao, Ph.D. thesis, University of Oxford, .[ ] Hartmut Klauck, Lower bounds for quantum commu-nication complexity, Proceedings of the nd IEEE Sym-posium on Foundations of Computer Science (FOCS’ ),pp. , , arXiv:quant-ph/ v .[ ] Scott Aaronson, Limitations of Quantum Advice and One-Way Communication, Proceedings of the th AnnualIEEE Conference on Computational Complexity (CCC’ ),pp. – , . arXiv:quant-ph/ v .[ ] M. Hayashi, K. Iwama, H. Nishimura, R. Raymond,S. Yamashita, Quantum Network Coding, arXiv:quant-ph/ .[ ] I. Kerenidis and R. de Wolf, Exponential Lower Boundfor -Query Locally Decodable Codes via a QuantumArgument, J. Comput. Syst. Sci., vol. , , pp. – , , arXiv:quant-ph/ v .[ ] S. Wehner and R. de Wolf, Improved Lower Boundsfor Locally Decodable Codes and Private Information Retrieval, Automata, Languages and Programming, pp. – , , arXiv:quant-ph/ v .[ ] Scott Aaronson, The Learnability of Quantum States, Proc.Roy. Soc. London Ser. A, vol. , no. , pp. – , , arXiv:quant-ph/ v .[ ] H.-W. Li, Z.-Q. Yin, Y.-C. Wu, X.-B. Zou, S. Wang, W.Chen, G.-C. Guo, and Z.-F. Han, Semi-device-independentrandom-number expansion without entanglement, Phys.Rev. A , ( ).[ ] M. Pawlowski and N. Brunner, Semi-device-independentsecurity of one-way quantum key distribution, Phys. Rev.A , (R) ( ).[ ] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner,and G. J. Pryde, Preparation Contextuality Powers Parity-Oblivious Multiplexing, Phys. Rev. Lett. , ( ).[ ] M. Banik, S. S. Bhattacharya, A. Mukherjee, A. Roy, A.Ambainis, and A. Rai, Limited preparation contextualityin quantum theory and its relation to the Cirel’son bound,Phys. Rev. A , (R) ( ).[ ] A. Ambainis, M. Banik, A. Chaturvedi, D. Kravchenko,and A. Rai, Parity Oblivious d-Level Random Ac-cess Codes and Class of Noncontextuality Inequalities,arXiv: .05490